Unique Continuation Property and Local Asymptotics of Solutions to Fractional Elliptic Equations

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UNIQUE CONTINUATION PROPERTY AND LOCAL ASYMPTOTICS OF

SOLUTIONS TO FRACTIONAL ELLIPTIC EQUATIONS

MOUHAMED MOUSTAPHA FALL AND VERONICA FELLI

Abstract. Asymptotics of solutions to fractional elliptic equations with Hardy type potentialsis studied in this paper. By using an Almgren type monotonicity formula, separation of variables,and blow-up arguments, we describe the exact behavior near the singularity of solutions to linearand semilinear fractional elliptic equations with a homogeneous singular potential related to thefractional Hardy inequality. As a consequence we obtain unique continuation properties forfractional elliptic equations.

1. Introduction

The purpose of the present paper is to describe the asymptotic behavior of solutions to thefollowing class of fractional elliptic semilinear equations with singular homogeneous potentials

(1) (−∆)su(x)− λ

|x|2s u(x) = h(x)u(x) + f(x, u(x)), in Ω,

where u ∈ Ds,2(RN ) (see definition below) and Ω ⊂ RN is a bounded domain containing the origin,

N > 2s, s ∈ (0, 1), λ < ΛN,s := 22sΓ2(N+2s

4

)

Γ2(N−2s

4

) ,(2)

h ∈ C1(Ω \ 0), |h(x)| + |x · ∇h(x)| 6 Ch|x|−2s+ε as |x| → 0,(3)

f ∈ C1(Ω× R), t 7→ F (x, t) ∈ C1(Ω× R),

|f(x, t)t|+ |f ′t(x, t)t

2|+ |∇xF (x, t) · x| 6 Cf |t|p for a.e. x ∈ Ω and all t ∈ R,(4)

where 2 < p 6 2∗(s) = 2NN−2s , F (x, t) =

∫ t0 f(x, r) dr, Cf , Ch, ε > 0 are positive constants inde-

pendent of x ∈ Ω and t ∈ R, ∇xF denotes the gradient of F with respect to the x variable, andf ′t(x, t) =

∂f∂t (x, t).

We recall that for any ϕ ∈ C∞c (RN ) and s ∈ (0, 1), the fractional Laplacian (−∆)sϕ is defined

as

(5) (−∆)sϕ(x) = C(N, s) P.V.

∫

RN

ϕ(x) − ϕ(y)

|x− y|N+2sdy = C(N, s) lim

ρ→0+

∫

|x−y|>ρ

ϕ(x) − ϕ(y)

|x− y|N+2sdy

where P.V. indicates that the integral is meant in the principal value sense and

C(N, s) = π−N2 22s

Γ(N+2s

2

)

Γ(2− s)s(1− s).

The Dirichlet form associated to (−∆)s on C∞c (RN ) is given by

(6) (u, v)Ds,2(RN ) =C(N, s)

2

∫

R2N

(u(x)− u(y))(v(x) − v(y))

|x− y|N+2sdx dy =

∫

RN

|ξ|2sv(ξ)u(ξ) dξ,

Date: Revised version, June 27, 2013.M. M. Fall is supported by the Alexander von Humboldt foundation. V. Felli was partially supported by the

PRIN2009 grant “Critical Point Theory and Perturbative Methods for Nonlinear Differential Equations”.2010 Mathematics Subject Classification. 35R11, 35B40, 35J60, 35J75.Keywords. Fractional elliptic equations, Caffarelli-Silvestre extension, Hardy inequality, Unique continuation

property.

1

http://arxiv.org/abs/1301.5119v3

2 MOUHAMED MOUSTAPHA FALL AND VERONICA FELLI

where u denotes the unitary Fourier transform of u. It defines a scalar product thanks to (8) or(9) below. From now on we define Ds,2(RN ) as the completion on C∞

c (RN ) with respect to thenorm induced by the scalar product (6).

By a weak solution to (1) we mean a function u ∈ Ds,2(RN ) such that

(7) (u, ϕ)Ds,2(RN ) =

∫

Ω

(λ

|x|2s u(x) + h(x)u(x) + f(x, u(x))

)ϕ(x) dx, for all ϕ ∈ C∞

c (Ω).

We notice that the right hand side of (7) is well defined in view of assumptions (3)–(4), theHardy-Littlewood-Sobolev inequality

(8) SN,s‖u‖2L2∗(s)(RN ) 6 ‖u‖2Ds,2(RN ),

and the following Hardy inequality, due to Herbst in [19] (see also [27]),

(9) ΛN,s

∫

RN

u2(x)

|x|2s dx 6

∫

RN

|ξ|2s|u(ξ)|2 dξ = ‖u‖2Ds,2(RN ), for all u ∈ Ds,2(RN ).

It should also be remarked that, in (7), we allow p = 2∗(s) and that u is not prescribed outside Ω.One of the aim of this paper is to give the precise behaviour of a solution u to (1). The rate and

the shape of u are given by the the eigenvalues and the eigenfunctions of the following eigenvalueproblem

− divSN (θ1−2s

1 ∇SNψ) = µ θ1−2s1 ψ, in SN+ ,

− limθ1→0+ θ1−2s1 ∇SNψ · e1 = κsλψ, on ∂SN+ ,

(10)

where

(11) κs =Γ(1− s)

22s−1Γ(s)

and

SN+ = (θ1, θ2, . . . , θN+1) ∈ S

N : θ1 > 0 =z|z| : z ∈ R

N+1, z · e1 > 0,

with e1 = (1, 0, . . . , 0); we refer to section 2.1 for a variational formulation of (10). From classicalspectral theory (see section 2.1 for the details) problem (10) admits a diverging sequence of realeigenvalues with finite multiplicity

µ1(λ) 6 µ2(λ) 6 · · · 6 µk(λ) 6 · · ·Moreover µ1(λ) > −

(N−2s

2

)2, see Lemma 2.2. We notice that if λ = 0 then µk(0) > 0 for all k.

Our first main result is the following theorem.

Theorem 1.1. Let u ∈ Ds,2(RN ) be a nontrivial solution to (1) in a bounded domain Ω ⊂ RN

containing the origin as in (7) with s, λ, h and f satisfying assumptions (2), (3) and (4). Thenthere exists an eigenvalue µk0(λ) of (10) and an eigenfunction ψ associated to µk0(λ) such that

(12) τ−2s−N

2 −√( 2s−N

2 )2+µk0

(λ)u(τx) → |x|−N−2s

2 +√( 2s−N

2 )2+µk0

(λ)ψ(0,

x

|x|)

as τ → 0+,

in C1,αloc (B

′1 \ 0) for some α ∈ (0, 1), where B′

1 := x ∈ RN : |x| < 1, and, in particular,

(13) τ−2s−N

2 −√( 2s−N

2 )2+µk0(λ)u (τθ′) → ψ (0, θ′) in C1,α(SN−1) as τ → 0+,

where SN−1 = ∂SN+ .

We should mention that the above result is stated in such form for the sake of simplicity. In fact,for the eigenfunction ψ in (13), we obtain precisely its components in any basis of the eigenspacecorresponding to µk0(λ), see Theorem 4.1 (below). We also remark that µ1(λ) < 0 for λ > 0 andit is determined implicitly by the usual Gamma function, see Proposition 2.3.

Analogous results were obtained in [12, 13, 14] for corresponding equations involving the Lapla-cian (i.e. in the case s = 1) and Hardy-type potentials for different kinds of problems: in [12, 14]for Schrodinger equations with electromagnetic potentials and in [13] for Schrodinger equationswith inverse square many-particle potentials.

As a particular case of Theorem 1.1, if λ = 0 we obtain that the convergence stated in 12 holdsin C1,α(B′

1).

FRACTIONAL ELLIPTIC EQUATIONS 3

Corollary 1.2. Let u ∈ Ds,2(RN ) be a nontrivial weak solution to

(14) (−∆)su(x) = h(x)u(x) + f(x, u(x)), in Ω,

in a bounded domain Ω ⊂ RN with s ∈ (0, 1), h ∈ C1(Ω), and f satisfying (4). Then, for everyx0 ∈ Ω, there exists an eigenvalue µk0 = µk0(0) of problem (10) with λ = 0 and an eigenfunctionψ associated to µk0 such that

(15) τ−2s−N

2 −√( 2s−N

2 )2+µk0u(x0 + τ(x − x0))

→ |x− x0|−N−2s

2 +√( 2s−N

2 )2+µk0ψ

(0,

x− x0|x− x0|

)as τ → 0+,

in C1,α(x ∈ RN : x− x0 ∈ B′1).

Our next result contains the so called strong unique continuation property which is a directconsequence of Theorem 1.1.

Theorem 1.3. Suppose that all the assumptions of Theorem 1.1 hold true. Let u be a solution to(1) in a bounded domain Ω ⊂ RN containing the origin. If u(x) = o(|x|n) = o(1)|x|n as |x| → 0for all n ∈ N, then u ≡ 0 in Ω.

From Theorem 1.2 a unique continuation principle related to sets of positive measures follows.

Theorem 1.4. Let u ∈ Ds,2(RN ) be a weak solution to (14) in a bounded domain Ω ⊂ RN withs ∈ (0, 1), h ∈ C1(Ω), and f satisfying (4). If u ≡ 0 on a set E ⊂ Ω of positive measure, thenu ≡ 0 in Ω.

An interesting application of Theorem 1.4 is that the nodal sets of eigenfunctions for the frac-tional laplacian operator have zero Lebesgue measure. We should point out that this was a keyassumption in [15], where the authors studied the existence of weak solutions to some non-localequations.

Recent research in the field of second order elliptic equations has devoted a great attention tothe problem of unique continuation property in the presence of singular lower order terms, see e.g.[11, 20, 22]. Two different kinds of approach have been developed to treat unique continuation:a first one is due to Carleman [5] and is based on weighted priori inequalities, whereas a secondone is due to Garofalo and Lin [17] and is based on local doubling properties proved by Almgrenmonotonicity formula. In the present paper we will follow the latter approach. Furthermore, in thespirit of [12, 13, 14], the combination of monotonicity methods with blow-up analysis will enableus to prove not only unique continuation but also the precise asymptotics of solutions stated inTheorem 1.1.

As far as unique continuation from sets of positive measures is concerned, we mention [6] whereit was proved for second order elliptic operators by combining strong unique continuation propertywith the De Giorgi inequality. Since the validity of a the De Giorgi type inequality for the fractionalproblem (or even its extension, see (17)) seems to be hard to prove, we will base the proof ofTheorem 1.4 directly on the asymptotic of solutions proved in Theorem 1.1.

To explain our argument of proving the asymptotic behavior, let us we write (7) as

(16) (−∆)su = G(x, u) in Ω.

The proof of our results is based on the study of the Almgren frequency function at the origin 0:“ratio of the local energy over mass near the origin”. Due to the non-locality of the Dirichlet formassociated to (−∆)s, it is not clear how to set up an Almgren’s type frequency function using thisenergy as in the local case s = 1. A way out for this difficulty is to use the Caffarelli-Silvestreextension [4] which can be seen as a local version of (16). The Caffarelli-Silvestre extension of asolution u to (16) is a function w defined on

RN+1+ = z = (t, x) : t ∈ (0,+∞), x ∈ R

Nsatisfying w = u on Ω and solving in some weak sense (see Section 2 for more details) the boundaryvalue problem

(17)

div(t1−2s∇w) = 0, in R

N+1+ ,

− limt→0+

t1−2s ∂w∂t = κsG(x,w), on Ω.


Here and in the following, we write z = (t, x) ∈ RN+1+ with x ∈ RN and t > 0, and we identify RN

with ∂RN+1+ , so that Ω is contained in ∂RN+1

+ .We then consider the Almgren’s frequency function

N (r) =D(r)

H(r)

where

D(r) =1

rN−2s

[ ∫

B+r

t1−2s|∇w|2 dt dx − κs

∫

B′

r

G(x,w)w dx

], H(r) =

1

rN+1−2s

∫

S+r

t1−2sw2 dS,

being

B+r = z = (t, x) ∈ R

N+1+ : |z| < r, B′

r := x ∈ RN : |x| < r,

S+r = z = (t, x) ∈ R

N+1+ : |z| = r,

and dS denoting the volume element on N -dimensional spheres. We mention that an Almgren’sfrequency function for degenerate elliptic equations of the form (17) was first formulated in [4,Section 6].

As a first but nontrivial step, we prove that limr→0 N (r) := γ exists and it is finite, seeLemma 3.15. Next, we make a blow-up analysis by zooming around the origin the solution w

normalized also by√H . More precisely, setting wτ (z) =

w(τz)√H(τ)

, we have that wτ converges, (in

some Holder and Sobolev spaces) to w solving the limiting equationdiv(t1−2s∇w) = 0, in B+

1 ,

− limt→0+

t1−2s ∂w∂t = κsλ

|z|2s w, on B′1.

To obtain this, the fact that h is negligible with respect to the Hardy potential and f is at mostcritical with respect to the Sobolev exponent (see assumptions (3) and (4)) plays a crucial role; werefer to Lemma 4.2 for more details.

The main point is that the Almgren’s frequency for w is N (r) = limτ→0 N (τr) = γ, i.e. Nis constant; hence w and ∇w · z

|z| are proportional on L2(S+r ; t

1−2s). As a consequence we ob-

tain w(z) = ϕ(|z|)ψ(z|z|). By separating variables in polar coordinates, we obtain that ψ is an

eigenfunction of (10) for some eigenvalue µk0(λ). By the method of variation of constants andthe fact that w has finite energy near the origin, we then prove that ϕ(r) is proportional to

rγ and that γ = 2s−N2 +

√(2s−N

2

)2+ µk0(λ). We finally complete the proof by showing that

limr→0 r−2γH(r) > 0, see Lemma 4.5.

We should mention that in the recent literature, a great attention has been addressed to non-local fractional diffusion and many papers have been devoted to the study of existence, non-existence, regularity and qualitative properties of solutions to elliptic equations associated tofractional Laplace type operators, see e.g. [2, 3, 4, 7, 9, 10, 24, 25] and references therein. Inparticular, semilinear fractional elliptic equations involving the Hardy potential were treated in[9], where some existence and nonexistence results were obtained from lower bounds of positivesolutions. Adapting the ideas in [12] in the nonlocal case of the present paper, we provide thebehavior (therefore regularity) of solutions at the singularity. In particular, the asymptotics wehave here show that the lower bound of positive solutions proved in [9] is sharp.

2. Preliminaries and notations

Notation. We list below some notation used throughout the paper.

- SN = z ∈ RN+1 : |z| = 1 is the unit N -dimensional sphere.

- SN+ = (θ1, θ2, . . . , θN+1) ∈ SN : θ1 > 0 := SN ∩ RN+1+ .

- dS denotes the volume element on N -dimensional spheres.- dS′ denotes the volume element on (N − 1)-dimensional spheres.


Let D1,2(RN+1+ ; t1−2s) be the completion of C∞

c (RN+1+ ) with respect to the norm

‖w‖D1,2(RN+1+ ;t1−2s) =

(∫

RN+1+

t1−2s|∇w(t, x)|2dt dx)1/2

.

We recall that there exists a well defined continuous trace map Tr : D1,2(RN+1+ ; t1−2s) → Ds,2(RN ),

see e.g. [2].

For every u ∈ Ds,2(RN ), let H(u) ∈ D1,2(RN+1+ ; t1−2s) be the unique solution to the minimiza-

tion problem

(18)

∫

RN+1+

t1−2s|∇H(u)|2 dt dx

= min

∫

RN+1+

t1−2s|∇v|2 dt dx : v ∈ D1,2(RN+1+ ; t1−2s), Tr(v) = u

.

By Caffarelli and Silvestre [4] we have that

(19)

∫

RN+1+

t1−2s∇H(u) · ∇ϕ dt dx = κs(u,Tr ϕ)Ds,2(RN ) for all ϕ ∈ D1,2(RN+1+ ; t1−2s),

where κs is defined in (11).We notice that combining (9), (18), (19) and (8), we obtain the following Hardy-trace inequality

κsΛN,s

∫

RN

(Tr v)2(x)

|x|2s dx 6

∫

RN+1+

t1−2s|∇v|2 dt dx, for all v ∈ D1,2(RN+1+ ; t1−2s)(20)

and also the Sobolev-trace inequality

κsSN,s‖Tr v‖2L2∗(s)(RN ) 6

∫

RN+1+

t1−2s|∇v|2 dt dx, for all v ∈ D1,2(RN+1+ ; t1−2s).(21)

2.1. Separation of variables in the extension operator. For every R > 0, we define the space

H1(B+R ; t

1−2s) as the completion of C∞(B+R ) with respect to the norm

‖w‖H1(B+R ;t1−2s) =

(∫

B+R

t1−2s(|∇w(t, x)|2 + w2(t, x)

)dt dx

)1/2.

By direct calculations, we can obtain the following lemma concerning separation of variables inthe extension operator.

Lemma 2.1. If v ∈ H1(B+R ; t

1−2s) is such that v(z) = f(r)ψ(θ) for a.e. z = (t, x) ∈ RN+1+ , with

r = |z| < R and θ = z|z| ∈ SN , then

div(t1−2s∇v(z)) = 1

rN(rN+1−2sf ′)′θ1−2s

1 ψ(θ) + r−1−2sf(r) divSN (θ1−2s1 ∇SNψ(θ))

in the distributional sense, where θ1 = tr = θ · e1 with e1 = (1, 0, . . . , 0), divSN (respectively ∇SN )

denotes the Riemannian divergence (respectively gradient) on the unit sphere SN endowed with the

standard metric.

Let us define H1(SN+ ; θ1−2s1 ) as the completion of C∞(SN+ ) with respect to the norm

‖ψ‖H1(SN+ ;θ1−2s1 ) =

(∫

SN+

θ1−2s1

(|∇SNψ(θ)|2 + ψ2(θ)

)dS

)1/2.

We also denote

L2(SN+ ; θ1−2s1 ) :=

ψ : SN+ → R measurable such that

∫SN+θ1−2s1 ψ2(θ) dS < +∞

.

The following trace inequality on the unit half-sphere SN+ holds.


Lemma 2.2. There exists a well defined continuous trace operator

H1(SN+ ; θ1−2s1 ) → L2(∂SN+ ) = L2(SN−1).

Moreover for every ψ ∈ H1(SN+ ; θ1−2s1 )

κsΛN,s

(∫

SN−1

|ψ(θ′)|2 dS′)

6

(N − 2s

2

)2 ∫

SN+

θ1−2s1 |ψ(θ)|2 dS +

∫

SN+

θ1−2s1 |∇SNψ(θ)|2 dS.

where dS′ denotes the volume element on the sphere SN−1 = ∂SN+ = (θ1, θ′) ∈ S

N+ : θ1 = 0.

Proof. Let ψ ∈ C∞(SN+ ) and f ∈ C∞c (0,+∞) with f 6= 0. Rewriting (20) for v(z) = f(r)ψ(θ),

r = |z|, θ = z|z| , we obtain that

κsΛN,s

(∫ +∞

0

rN−1−2s f2(r) dr

)(∫

SN−1

|ψ(0, θ′)|2 dS′)

6

(∫ +∞

0

rN+1−2s|f ′(r)|2 dr)(∫

SN+

θ1−2s1 |ψ(θ)|2 dS

)

+

(∫ +∞

0

rN−1−2sf2(r) dr

)(∫

SN+

θ1−2s1 |∇SN+

ψ(θ)|2 dS),

and hence, by optimality of the classical Hardy constant, see [18, Theorem 330],

κsΛN,s

(∫

SN−1

|ψ(0, θ′)|2 dS′)

6

(∫

SN+

θ1−2s1 |ψ(θ)|2 dS

)inf

f∈C∞

c (0,+∞)

∫ +∞0

rN+1−2s|f ′(r)|2 dr∫ +∞0 rN−1−2s f2(r) dr

+

∫

SN+

θ1−2s1 |∇SN+

ψ(θ)|2 dS

=(N − 2s

2

)2 ∫

SN+

θ1−2s1 |ψ(θ)|2 dS +

∫

SN+

θ1−2s1 |∇SN+

ψ(θ)|2 dS.

By density of C∞(SN+ ) in H1(SN+ ; θ1−2s1 ), we obtain the conclusion.

In view of Lemma 2.1, in order to construct an orthonormal basis of L2(SN+ ; θ1−2s1 ) for expanding

solutions to (17) in Fourier series, we are naturally lead to consider the eigenvalue problem (10),which admits the following variational formulation: we say that µ ∈ R is an eigenvalue of problem(10) if there exists ψ ∈ H1(SN+ ; θ1−2s

1 ) \ 0 (called eigenfunction) such that

Q(ψ, υ) = µ

∫

SN+

θ1−2s1 ψ(θ)υ(θ) dS, for all υ ∈ H1(SN+ ; θ1−2s

1 ),

where

Q : H1(SN+ ; θ1−2s1 )×H1(SN+ ; θ1−2s

1 ) → R,

Q(ψ, υ) =

∫

SN+

θ1−2s1 ∇SNψ(θ) · ∇υ(θ) dS − λκs

∫

SN−1

T ψ(θ′)T υ(θ′) dS′.

By Lemma 2.2 the bilinear form Q is continuous and weakly coercive on H1(SN+ ; θ1−2s1 ). Moreover

the belonging of the weight t1−2s to the second Muckenhoupt class ensures that the embeddingH1(SN+ ; θ1−2s

1 ) →→ L2(SN+ ; θ1−2s1 ) is compact (see [8] for weighted embeddings with Muckenhoupt

A2 weights). Then, from classical spectral theory (see e.g. [23, Theorem 6.16]), problem (10) admitsa diverging sequence of real eigenvalues with finite multiplicity µ1(λ) 6 µ2(λ) 6 · · · 6 µk(λ) 6 · · ·the first of which admits the variational characterization

(22) µ1(λ) = minψ∈H1(SN+ ;θ1−2s

1 )\0

Q(ψ, ψ)∫SN+θ1−2s1 ψ2(θ) dS

.

Furthermore, in view of Lemma 2.2, we have that

(23) µ1(λ) > −(N − 2s

2

)2.


To each k > 1, we associate an L2(SN+ ; θ1−2s1 )-normalized eigenfunction ψk ∈ H1(SN+ ; θ1−2s

1 ) \ 0corresponding to the k-th eigenvalue µk(λ), i.e. satisfying

(24) Q(ψk, υ) = µk(λ)

∫

SN+

θ1−2s1 ψk(θ)υ(θ) dS, for all υ ∈ H1(SN+ ; θ1−2s

1 ).

In the enumeration µ1(λ) 6 µ2(λ) 6 · · · 6 µk(λ) 6 · · · we repeat each eigenvalue as many timesas its multiplicity; thus exactly one eigenfunction ψk corresponds to each index k ∈ N, k > 1. Wecan choose the functions ψk in such a way that they form an orthonormal basis of L2(SN+ ; θ1−2s

1 ).We can also determine µ1(λ) for λ ∈ (0,ΛN,s), where ΛN,s is the fractional Hardy constant

defined in (2).

Proposition 2.3. For every α ∈(0, N−2s

2

), we define

λ(α) = 22sΓ(N+2s+2α

4

)

Γ(N−2s−2α

4

) Γ(N+2s−2α

4

)

Γ(N−2s+2α

4

) .

Then the mapping α 7→ λ(α) is continuous and decreasing. In addition we have that

µ1(λ(α)) = α2 −(N − 2s

2

)2

for all α ∈(0,N − 2s

2

).

Proof. It was proved in [9, Lemma 3.1] that, for every α ∈(0, N−2s

2

), there exists a positive

continuous function Φα : RN+1+ → R such that

(25)

div(t1−2s∇Φα) = 0 in RN+1+

Φα = |x| 2s−N2 +α on RN \ 0

−t1−2s ∂Φα

∂t = κsλ(α)|x|−2s Φα on RN \ 0,

Moreover Φα ∈ H1(B+R ; t

1−2s) for every R > 0 and Φα is scale invariant, i.e.

Φα(τz) = τ2s−N

2 +αΦα(z), for all τ > 0,

thus implying that, for all z ∈ RN+1+ ,

(26) c1|z|2s−N

2 +α 6 Φα(z) 6 c2|z|2s−N

2 +α,

for some positive constants c1, c2. It is also known (see for instance [16]) that the map α 7→ λ(α)is continuous and monotone decreasing.

We write

Φα(z) =

∞∑

k=1

Φkα(|z|)ψk(z/|z|), Φkα(r) =

∫

SN+

ψk(θ)Φα(rθ)dS.

In particular, since ψ1 > 0, by (26) we have, for every r ∈ (0, R),

(27) c′1|r|2s−N

2 +α 6 Φ1α(r) 6 c′2|r|

2s−N2 +α,

for some positive constants c′1, c′2. Using (25), we have, weakly, for every k > 1 and r ∈ (0, R),

1rN

(rN+1−2s(Φkα)

′)′θ1−2s1 ψk(θ) + r−1−2sΦkα divSN (θ

1−2s1 ∇SNψk(θ)) = 0,

−Φkα limθ1→0+ θ1−2s1 ∇SNψk(θ) · e1 = κs λ(α)ψk(0, θ

′)Φkα.

Testing the above equation with ψ1 > 0 and using also the fact that Φ1α > 0, we obtain

(Φ1α)

′′ +N + 1− 2s

r(Φ1

α)′ − µ1(λ(α))

r2Φ1α = 0

and hence Φ1α(r) is of the form

Φ1α(r) = c3r

σ+

+ c4rσ−

for some c3, c4 ∈ R, where

σ+ = −N − 2s

2+

√(N − 2s

2

)2+ µ1(λ(α)) and σ− = −N − 2s

2−√(

N − 2s

2

)2+ µ1(λ(α)).


Since the function |z|σ−

k0ψ1(z|z|) /∈ H1(B+

1 ; t1−2s), we deduce that c4 = 0 and thus for every

r ∈ (0, R)

Φ1α(r) = c3r

σ+

.

This together with (27) implies that

µ1(λ(α)) = α2 −(N − 2s

2

)2

for all α ∈(0,N − 2s

2

),

as claimed.

2.2. Hardy type inequalities. From well-known weighted embedding inequalities and the factthat the weight t1−2s belongs to the second Muckenhoupt class (see e.g. [8]), the embeddingH1(B+

r ; t1−2s) → L2(B+

r ; t1−2s) is compact. It can be also proved that both the trace operators

(28) H1(B+r ; t

1−2s) →→ L2(S+r ; θ

1−2s1 ),

(29) H1(B+r ; t

1−2s) →→ L2(B′r)

are well defined and compact.

For sake of simplicity, in the following of this paper, we will often denote the trace of a functionwith the same letter as the function itself.

The following Hardy type inequality with boundary terms holds.

Lemma 2.4. For all r > 0 and w ∈ H1(B+r ; t

1−2s), the following inequality holds(N − 2s

2

)2 ∫

B+r

t1−2sw2(z)

|z|2 dz 6

∫

B+r

t1−2s

(∇w(z) · z|z|

)2dz +

(N − 2s

2r

)∫

S+r

t1−2sw2dS.

Proof. By scaling, it is enough to prove the stated inequality for r = 1. Let

V (z) = |z| 2s−N2 , z ∈ R

N+1+ \ 0.

We notice that V satisfies

(30) − div(t1−2s∇V ) =

(N − 2s

2

)2t1−2s|z|−2V (z) in R

N+1+ \ 0.

Hence, letting w ∈ C∞(B+1 ), multiplying (30) with w2

V , and integrating overB+1 \B+

δ with δ ∈ (0, 1),we obtain(N − 2s

2

)2 ∫

B+1 \B+

δ

t1−2sw2(z)

|z|2 dz

=

∫

B+1 \B+

δ

t1−2s∇V (z) · ∇(w2

V

)(z) dz −

∫

S+1

t1−2s(∇V · ν)w2

VdS +

∫

S+δ

t1−2s(∇V · ν)w2

VdS

= −(N − 2s)

∫

B+1 \B+

δ

t1−2s w

|z|(∇w · z|z|

)dz −

(N − 2s

2

)2 ∫

B+1 \B+

δ

t1−2s w2

|z|2 dz

+N − 2s

2

∫

S+1

t1−2sw2dS − N − 2s

2

1

δ

∫

S+δ

t1−2sw2dS

where ν(z) = z|z| . Since, by Schwarz’s inequality

(31) − (N − 2s)

∫

B+1 \B+

δ

t1−2s w

|z|(∇w · z|z|

)dz

6

(N − 2s

2

)2 ∫

B+1 \B+

δ

t1−2s w2

|z|2 dz +∫

B+1 \B+

δ

t1−2s(∇w · z|z|

)2dz

and1

δ

∫

S+δ

t1−2sw2dS = O(δN−2s) = o(1) for δ → 0+,


letting δ → 0 we obtain the stated inequality for r = 1 and w ∈ C∞(B+1 ). The conclusion follows

by density of C∞(B+1 ) in H

1(B+1 ; t

1−2s).

Lemma 2.5. For every r > 0 and w ∈ H1(B+r ; t

1−2s), the following inequality holds

κsΛN,s

∫

B′

r

w2

|x|2s dx 6

(N − 2s

2r

)∫

S+r

t1−2sw2dS +

∫

B+r

t1−2s|∇w|2dz.

Proof. Let w ∈ C∞(B+r ). Then, passing to polar coordinates and using Lemmas 2.2 and 2.4,

we obtain

κsΛN,s

∫

B′

r

w2(0, x)

|x|2s dx = κsΛN,s

∫ r

0

ρN−1−2s

(∫

SN−1

w2(0, ρθ′) dS′)dρ

6

∫ r

0

ρN−1−2s

((N − 2s

2

)2 ∫

SN+

θ1−2s1 |w(ρθ)|2 dS +

∫

SN+

θ1−2s1 |∇SNw(ρθ)|2 dS

)dρ

=(N − 2s

2

)2 ∫

B+r

t1−2s w2

|z|2 dz +∫ r

0

ρN−1−2s

(∫

SN+

θ1−2s1 |∇SNw(ρθ)|2 dS

)dρ

6

(N − 2s

2r

)∫

S+r

t1−2sw2dS

+

∫ r

0

ρN+1−2s

(∫

SN+

θ1−2s1

(1

ρ2|∇SNw(ρθ)|2 +

∣∣∣∂w∂ρ

(ρθ)∣∣∣2)dS

)dρ

=

(N − 2s

2r

)∫

S+r

t1−2sw2dS +

∫

B+r

t1−2s|∇w|2dz.

The conclusion follows by density of C∞(B+r ) in H1(B+

r ; t1−2s).

The following Sobolev type inequality with boundary terms holds.

Lemma 2.6. There exists SN,s > 0 such that, for all r > 0 and w ∈ H1(B+r ; t

1−2s),(∫

B′

r

|w|2∗(s) dx) 2

2∗(s)

6 SN,S

[N − 2s

2r

∫

S+r

t1−2sw2dS +

∫

B+r

t1−2s|∇w|2dz].

Proof. By scaling, it is enough to prove the statement for r = 1. Let w ∈ C∞(B+1 ) and denote

by w its fractional Kelvin transform defined as w(z) = |z|−(N−2s)w(z

|z|2). Some computations (see

also [10]) show that∫

B+1

t1−2s|∇w|2dz + (N − 2s)

∫

S+1

t1−2sw2dS =

∫

RN+1+ \B+

1

t1−2s|∇w|2dz,(32)

∫

B′

1

|w|2∗(s)dx =

∫

RN\B′

1

|w|2∗dx.(33)

In particular the function

v(z) =

w(z), if z ∈ B+

1 ,

w(z), if z ∈ RN+1+ \B+

1

belongs to D1,2(RN+1+ ; t1−2s), so that, by (21) we have

(34)

(∫

RN

|v|2∗(s) dx) 2

2∗(s)

61

SN,sκs

∫

RN+1+

t1−2s|∇v|2dz.

From (32), (33), and (34), it follows that(2

∫

B′

1

|w|2∗(s)dx) 2

2∗(s)

62

SN,sκs

[∫

B+1

t1−2s|∇w|2dz + N − 2s

2

∫

S+1

t1−2sw2dS

]

which, by density, yields the conclusion.

Combining Lemma 2.5 and Lemma 2.6 the following corollary follows.


Corollary 2.7. For all r > 0 and w ∈ H1(B+r ; t

1−2s), the following inequalities hold

(35)

∫

B+r

t1−2s|∇w|2dz − κsλ

∫

B′

r

w2

|x|2s dx+N − 2s

2r

∫

S+r

t1−2sw2dS

> κs(ΛN,s − λ)

∫

B′

r

w2

|x|2s dx

and

(36)

∫

B+r


∫

B′

r

w2

|x|2s dx+N − 2s

2r

∫

S+r

t1−2sw2dS

>ΛN,s − λ

(1 + ΛN,s)SN,s

(∫

B′

r

|w|2∗(s) dx) 2

2∗(s)

.

3. The Almgren type frequency function

Let R > 0 be such that B′R ⊂⊂ Ω and w ∈ H1(B+

R ; t1−2s) be a nontrivial solution to

(37)

div(t1−2s∇w) = 0, in B+

R ,

− limt→0+ t1−2s ∂w

∂t (t, x) = κs

(λ

|x|2sw + hw + f(x,w)), on B′

R,

in a weak sense, i.e., for all ϕ ∈ C∞c (B+

R ∪B′R), we have that

(38)

∫

RN+1+

t1−2s∇w · ∇ϕdt dx = κs

∫

B′

R

(λ

|x|2sw + hw + f(x,w)

)ϕdx,

with s, λ, h, f as in assumptions (2), (3), and (4).For every r ∈ (0, R] we define

(39) D(r) =1

rN−2s

[∫

B+r

t1−2s|∇w|2 dt dx− κs

∫

B′

r

(λ

|x|2sw2 + hw2 + f(x,w)

)dx

]

and

(40) H(r) =1

rN+1−2s

∫

S+r

t1−2sw2 dS =

∫

SN+

θ1−2s1 w2(rθ) dS.

The main result of this section is the existence of the limit as r → 0+ of the Almgren’s frequencyfunction (see [17] and [1]) associated to w

(41) N (r) =D(r)

H(r)=

r

[∫

B+r


∫

B′

r

(λ

|x|2sw2 + h(x)w2 + f(x,w)w

)dx

]

∫

S+r

t1−2sw2 dS

.

We notice that, by Lemma 2.5, w(0, ·) ∈ L2(B′R; |x|−2s) and so the L1(0, R)-function

r 7→∫

S+r

t1−2s|∇w|2 dS, respectively r 7→∫

∂B′

r

w2

|x|2s dS′,

is the weak derivative of the W 1,1(0, R)-function

r →∫

B+r

t1−2s|∇w|2 dz, respectively r 7→∫

B′

r

w2

|x|2s dx.

In particular, for a.e. r ∈ (0, R), ∂w∂ν ∈ L2(S+r ; t

1−2s), where ν = ν(z) = z|z| .

Next we observe the following integration by parts.

Lemma 3.1. For a.e. r ∈ (0, R) and every ϕ ∈ C∞(B+r )

∫

B+r

t1−2s∇w · ∇ϕ dz =∫

S+r

t1−2s ∂w

∂νϕ dS + κs

∫

B′

r

(λ


)ϕ dx.


Proof. It follows by testing (38) with ϕ(z)ηn(|z|) where ηn(ρ) = 1 if ρ < r − 1n , ηn(r) = 0 if

ρ > r, η(ρ) = n(r − ρ) if r − 1n 6 ρ 6 r, passing to the limit, and noticing that a.e. r ∈ (0, R) is a

Lebesgue point for the L1(0, R)- function r 7→∫S+rt1−2s ∂w

∂ν ϕ dS.

3.1. Regularity estimates and Pohozaev-type identity. In this section, we will prove localregularity estimates for a general class of fractional elliptic equations in Lemma 3.3 below. Thisestimate will be useful for the blow-up analysis and also for establishing the Pohozaev identity inTheorem 3.7 below which is crucial in this paper. The proof of Lemma 3.3 uses mainly a result ofJin, Li and Xiong in [21] that we state here for sake of completeness.

Proposition 3.2 ([21] Proposition 2.4). Let v ∈ H1(B+1 ; t1−2s) be a weak solution to

div(t1−2s∇v) = 0, in B+

1

− limt→0+ t1−2svt = c(x)v + b(x), on B′

1,

with c, b ∈ Lp(B′1) for some p > N

2s . Then v ∈ L∞loc

(B+

1 ∪B′1

)and there exists C > 0 depending

only on N, s, p, ‖c‖Lp(B′

1)such that

‖v‖L∞

([0,1/2)×B′

1/2

) 6 C(‖v‖H1(B+

1 ;t1−2s) + ‖b‖Lp(B′

1)

).

Also v ∈ C0,αloc (B+

r ∪B′r)) for some α ∈ (0, 1) depending only on N, s, p, ‖c‖Lp(B′

1). In addition we

have

‖v‖C0,α

([0,1/2)×B′

1/2

) 6 C(‖v‖L∞([0,1)×B′

1)+ ‖b‖Lp(B′

1)

).

We now state the following technical but crucial result.

Lemma 3.3. (i) Let r > 0 and V ∈ Lq(B′r) for some q > N

2s . For every t0, r0 > 0 such that[0, t0) × B′

r0 ⋐ B+r ∪ B′

r there exist positive constants A1 > 0, α ∈ (0, 1) depending on

t0, r0, r, ‖V ‖Lq(B′

r)such that for every v ∈ H1(B+

r ; t1−2s) solving

div(t1−2s∇v) = 0, in B+

r

− limt→0+ t1−2svt = V (x)v, on B′

r,

we have that v ∈ C0,α([0, t0)×B′r0) and

(42) ‖v‖C0,α([0,t0)×B′

r0) 6 A1‖v‖H1(B+

r ;t1−2s).

(ii) Let v ∈ H1(B+r ; t

1−2s) ∩ C0,α(B+r ) for some α ∈ (0, 1) be a function satisfying

div(t1−2s∇v) = 0, in B+

r

− limt→0+ t1−2svt = g(x, v), on B′

r,

where g ∈ C1(B′r × R) and |g(x, ρ)| 6 c(|ρ| + |ρ|p−1) for some 2 < p 6 2∗(s) = 2N

N−2s , c > 0,

and every x ∈ B′r and ρ ∈ R. Let t0, r0 > 0 such that [0, t0) × B′

r0 ⋐ B+r . Then there

exist positive constants A2, β depending only on N , p, s, c, r, r0, t0, ‖v‖H1(B+r ;t1−2s) and

‖g‖C1(B′

r0×[0,A3]) where A3 = ‖v‖C0,α([0,t0)×B′

r0), with β ∈ (0, 1), such that

(43) ‖∇xv‖C0,β([0,t0)×B′

r0) 6 A2,

(44) ‖t1−2svt‖C0,β([0,t0)×B′

r0) 6 A2.

Remark 3.4. The dependence of the constant A2 in Lemma 3.3 on ‖v‖H1(B+r ;t1−2s) is continuous;

in particular we can take the same A2 for a family of solutions which are uniformly bounded inH1(B+

r ; t1−2s) ∩ C0,α(B+

r ).

Proof. Part (i) and (42) follows from Proposition 3.2.

To prove (ii), for h ∈ RN with |h| << 1, we set vh(t, x) = v(t,x+h)−v(t,x)|h| for every x ∈ B′

r0/2.

Then we have div(t1−2s∇vh) = 0, in B+

r0/2

− limt→0+ t1−2svht = ch(x)v

h + bh, on B′r0/2

,


where

ch(x) =g(x, v(0, x + h))− g(x, v(0, x))

v(0, x+ h)− v(0, x)χv(0,x+h) 6=v(0,x)(x)

with χA being the characteristic function of a set A and

bh(x) =g(x+ h, v(0, x+ h))− g(x, v(0, x+ h))

|h| .

Let A3 = ‖v‖C0,α([0,t0)×B′

r0). Then we have

‖ch‖L∞(B′

r0/4) + ‖bh‖L∞(B′

r0/4) 6 ‖gρ‖L∞(B′

r0/2×[0,A3]) + ‖∇xg‖L∞(B′

r0/2×[0,A3]),

for every small h. Applying once again Proposition 3.2,

‖vh‖C0,α([0,t0/8)×B′

r0/8) 6 C(‖vh‖L∞([0,t0/4)×B′

r0/4) + ‖bh‖L∞(B′

r0/2))

6 C‖vh‖L2([0,t0/2)×B′

r0/2;t1−2s) + C‖∇g‖L∞(B′

r0/2×[0,A3])

6 C‖∇v‖L2([0,t0)×B′

r0;t1−2s) + C‖∇g‖L∞(B′

r0/2×[0,A3])

6 A2

for every small h. By the Arzela-Ascoli Theorem, passing to the limit as |h| → 0, we conclude thatz 7→ ∇xv(z) ∈ C([0, t0/8)×B′

r0/8) and estimate (43) holds for all β ∈ (0, α).

Since the map x 7→ g(x, v(0, x)) ∈ C0,β(B′r0/8

), estimate (44) follows from [[3], Lemma 4.5].

Because of the presence of the Hardy potential |x|−2s, we cannot expect the solution w of (37)

to be Lqloc near the origin for every q, but we can expect w to be in a space better than L2∗(s)loc .

This is what we will prove in the next result.

Lemma 3.5. Let w be a solution to (37) in the sense of (38). Then there exist p0 > 2∗(s) andR0 ∈ (0, R) such that w ∈ Lp0(B+

R0).

Proof. By (3), there are δ > 0 and Rδ ∈ (0, R) such that

(45) (λ+ |x|2s|h(x)|) 6 λ+ δ < ΛN,s, for all x ∈ B′Rδ.

Let β > 1. For all L > 0, we define FL(τ) = |τ |β if |τ | < L and FL(τ) = βLβ−1|τ | + (1 − β)Lβ if|τ | > L. Put G=

1βFLF

′L. It is easy to verify that, for all τ ∈ R,

(46) τGL(τ) 6 τ2G′L(τ), τGL(τ) 6 (FL(τ))

2, (F ′L(τ))

2 6 βG′L(τ).

Let η ∈ C∞c (B+

Rδ∪ B′

Rδ) be a radial cut-off function such that η ≡ 1 in B+

Rδ/2. It is clear that

ζ := η2GL(w) ∈ H1(B+R ; t

1−2s) and FL(w) ∈ H1(B+R ; t

1−2s). Using ζ as a test in (37), from (45)and integration by parts, we have that∫

B+Rδ

t1−2sη2|∇w|2G′L(w)dtdx − κs(λ+ δ)

∫

B′

Rδ

|x|−2sη2wGL(w)dx

6 −2

∫

B+Rδ

t1−2sη∇w · ∇ηGL(w)dtdx

+ c

∫

B′

Rδ

η2wGL(w)dx + c

∫

B′

Rδ

η2|w|2∗(s)−2wGL(w)dx,

for some positive c > 0 depending only on Cf , s, p,N . By Young’s inequality and (46), we havethat, for every σ > 0,∣∣∣∣2(η∇w) · (w∇η)

GL(w)

w

∣∣∣∣ 6σ

2η2|∇w|2G′

L(w) +2

σ|∇η|2(FL(w))2,

hence we obtain that(1− σ

2

) ∫

B+Rδ

t1−2sη2|∇w|2G′L(w) dt dx − κs(λ + δ)

∫

B′

Rδ

|x|−2sη2wGL(w) dx

6 c

∫

B′

Rδ

η2(|w|2∗(s)−2 + 1)wGL(w) dx +2

σ

∫

B+Rδ

t1−2s|∇η|2ψ2dt dx,


where we have set ψ = FL(w). Therefore by (46)

1

β

(1− σ

2

)∫

B+Rδ

t1−2sη2|∇ψ|2dt dx− κs(λ+ δ)

∫

B′

Rδ

|x|−2s(ηψ)2dx

6 c

∫

B′

Rδ

(|w|2∗(s)−2 + 1)(ηψ)2 dx +2

σ

∫

B+Rδ

t1−2s|∇η|2ψ2dt dx.

Since |∇(ηψ)|2 6 (1 + σ)η2|∇ψ|2 + (1 + 1σ )ψ

2|∇η|2, we have that

1

β(1 + σ)

(1− σ

2

)∫

B+Rδ

t1−2s|∇(ηψ)|2dt dx− κs(λ+ δ)

∫

B′

Rδ

|x|−2s(ηψ)2dx

6 c

∫

B′

Rδ

(|w|2∗(s)−2 + 1)(ηψ)2 dx+ C(c, β, σ,Rδ)

∫

B+Rδ

t1−2sψ2dt dx

for some positive C(c, β, σ,Rδ) > 0 depending only on c, σ, Rδ, and β. By Holder inequality andLemma 2.6, we have that∫

B′

Rδ

(|w|2∗(s)−2 + 1)(ηψ)2 dx

6 SN,s

(∫

B′

Rδ

|w|2∗(s)dx)2∗(s)−2

2∗(s)

+ |B′Rδ

| 2sN∫

B+Rδ

t1−2s|∇(ηψ)|2dt dx.

We deduce that

(47) A

∫

B+Rδ

t1−2s|∇(ηψ)|2dt dx − κs(λ+ δ)

∫

B′

Rδ

|x|−2s(ηψ)2dx 6 const

∫

B+Rδ

t1−2sψ2dtdx,

for some positive const > 0 depending only on Cf , s, p,N, β, σ,Rδ, where

A =1

(1 + σ)β

(1− σ

2

)− cSN,s

(∫

B′

Rδ

|w|2∗(s)dx)2∗(s)−2

2∗(s)

+ |B′Rδ

| 2sN .

From Hardy inequality (20), we have that

A

∫

B+Rδ

t1−2s|∇(ηψ)|2dt dx − κs(λ+ δ)

∫

B′

Rδ

|x|−2s(ηψ)2dx

>

(A− λ+ δ

ΛN,s

)∫

B+Rδ

t1−2s|∇(ηψ)|2dt dx,

and, by (45), we can choose β sufficiently close to 1 and σ,Rδ sufficiently small such that

A− λ+ δ

ΛN,s> 0.

Hence we have that

C

∫

B+Rδ

t1−2s|∇(ηψ)|2dt dx 6

∫

B+Rδ

t1−2sψ2dtdx,

for some constant C > 0 depending on f, h, s, p,N, β, ε, w, λ, δ. From Lemma 2.6 it follows that

CS−1N,s

(∫

B′

Rδ

|ηFL(w)|2∗(s)dx

) 22∗(s)

6

∫

B+Rδ

t1−2s|w|2βdtdx for all L > 0.

Hence by taking the limit as L→ +∞

(48) CS−1N,s

(∫

B′

Rδ/2

|w|β2∗(s)dx) 2

2∗(s)

6

∫

B+Rδ

t1−2s|w|2βdtdx.

The conclusion follows since β > 1 and H1(B+R ; t

1−2s) → Lq(B+R ; t

1−2s) for some q > 2, see forinstance [[8], Theorem 1.2].


Remark 3.6. From Lemma 3.5, we deduce that, if w ∈ H1(B+R ; t

1−2s) is a weak solution to (37),

then λ|x|2s + h + f(x,w)

w ∈ Lqloc(B′R \ 0) for some N

2s < q 6p0p−2 and hence, from Lemma 3.3, we

conclude that w ∈ C0,αloc (B

+r \ 0), ∇xv ∈ C0,β

loc (B+r \ 0), and t1−2svt ∈ C0,β

loc (B+r \ 0) for all

r ∈ (0, R) and some 0 < β < α < 1.

We next prove the following Pohozaev-type identity which will be used to compute D′ (see Lemma3.9 below) and therefore N ′.

Theorem 3.7. Let w solves (38). Then for a.e. r ∈ (0, R) there holds

(49) − N − 2s

2

[ ∫

B+r


∫

B′

r

w2

|x|2s dx]

+r

2

[ ∫

S+r

t1−2s|∇w|2dS − κsλ

∫

∂B′

r

w2

|x|2s dS′]

= r

∫

S+r

t1−2s

∣∣∣∣∂w

∂ν

∣∣∣∣2

dS − κs2

∫

B′

r

(Nh+∇h · x)w2 dx+rκs2

∫

∂B′

r

hw2 dS′

+ rκs

∫

∂B′

r

F (x,w) dS′ − κs

∫

B′

r

[∇xF (x,w) · x+NF (x,w)] dx

and

(50)

∫

B+r


∫

B′

r

w2

|x|2s dx

=

∫

S+r

t1−2s ∂w

∂νw dS + κs

∫

B′

r

(hw2 + f(x,w)w

)dx.

Proof. We write our problem in the formdiv(t1−2s∇w) = 0, in B+

R ,

− limt→0+ t1−2swt = G(x,w), on B′

R,

where G ∈ C1(B′R \ 0 × R), G(x, ) = κs

(λ

|x|2s + h(x)+ f(x, )).

We have, on B+R , the formula

(51) div

(1

2t1−2s|∇w|2z − t1−2s(z · ∇w)∇w

)=N − 2s

2t1−2s|∇w|2 − (z · ∇w) div(t1−2s∇w).

Let ρ < r < R. Now we integrate by parts over the set Oδ := (B+r \ B+

ρ ) ∩ (t, x), t > δ withδ > 0. We have

N − 2s

2

∫

Oδ

t1−2s|∇w(z)|2dz = −1

2δ2−2s

∫

B′√r2−δ2

\B′√ρ2−δ2

|∇w|2(δ, x)dx

+ δ2−2s

∫

B′√r2−δ2

\B′√ρ2−δ2

|wt|2(δ, x)dx

+r

2

∫

S+r ∩t>δ

t1−2s|∇w|2dS − r

∫

S+r ∩t>δ

t1−2s

∣∣∣∣∂w

∂ν

∣∣∣∣2

dS

− ρ

2

∫

S+ρ ∩t>δ

t1−2s|∇w|2dS + ρ

∫

S+ρ ∩t>δ

t1−2s

∣∣∣∣∂w

∂ν

∣∣∣∣2

dS

+

∫

B′√r2−δ2

\B′√ρ2−δ2

(x · ∇xw(δ, x)) δ1−2swt(δ, x) dx.

We now claim that there exists a sequence δn → 0 such that

limn→∞

[1

2δ2−2sn

∫

B′

r

|∇w|2(δn, x)dx + δ2−2sn

∫

B′

r

|wt|2(δn, x)dx]= 0.


If no such sequence exists, we would have

lim infδ→0

[1

2δ2−2s

∫

B′

r

|∇w|2(δ, x)dx + δ2−2s

∫

B′

r

|wt|2(δ, x)dx]> C > 0

and thus there exists δ0 > 0 such that

1

2δ2−2s

∫

B′

r

|∇w|2(δ, x)dx+ δ2−2s

∫

B′

r

|wt|2(δ, x)dx >C

2for all δ ∈ (0, δ0).

It follows that

1

2δ1−2s

∫

B′

r

|∇w|2(δ, x)dx + δ1−2s

∫

B′

r

|wt|2(δ, x)dx >C

2δfor all δ ∈ (0, δ0)

and so integrating the above inequality on (0, δ0) we contradict the fact that w ∈ H1(B+R ; t

1−2s).Next, from the Dominated Convergence Theorem, Lemma 3.3 and Remark 3.6, we have that

limδ→0

∫

B′√r2−δ2

\B′√ρ2−δ2

(x · ∇xw(δ, x)) δ1−2swt(δ, x) dx = −

∫

B′

r\B′

ρ

(x · ∇xw)G(x,w) dx.

We conclude that (replacing Oδ with Oδn , for a sequence δn → 0) that

(52)N − 2s

2

∫

B+r \B+

ρ

t1−2s|∇w(z)|2dz =r

2

∫

S+r

t1−2s|∇w|2dS − r

∫

S+r

t1−2s

∣∣∣∣∂w

∂ν

∣∣∣∣2

dS

− ρ

2

∫

S+ρ

t1−2s|∇w|2dS − ρ

∫

S+ρ

t1−2s

∣∣∣∣∂w

∂ν

∣∣∣∣2

dS −∫

B′

r\B′

ρ

(x · ∇xw)G(x,w) dx.

Furthermore, integration by parts yields∫

B′

r\B′

ρ

(x · ∇xw)G(x,w) dx = −N − 2s

2κsλ

∫

B′

r\B′

ρ

w2

|x|2s dx(53)

− κs2

∫

B′

r\B′

ρ

(Nh(x) +∇h(x) · x)w2 dx+ κsλr

2

∫

∂B′

r

w2

|x|2s dS′

+rκs2

∫

∂B′

r

h(x)w2 dS′ − κsλρ

2

∫

∂B′

ρ

w2

|x|2s dS′ − ρκs

2

∫

∂B′

ρ

h(x)w2 dS′

− κs

∫

B′

r\B′

ρ

[∇xF (x,w) · x+NF (x,w)] dx

+ rκs

∫

∂B′

r

F (x,w) dS′ − ρκs

∫

∂B′

ρ

F (x,w) dS′.

Since w ∈ H1(B+R ; t

1−2s), in view of Lemma 2.5 and (8), there exists a sequence ρn → 0 such that

limn→∞

ρn

[∫

S+ρn

t1−2s|∇w|2dS +

∫

∂B′

ρn

w2

|x|2s dS′ +

∫

∂B′

ρn

|F (x,w)| dS′]= 0.

Hence, taking ρ = ρn and letting n→ ∞ in (52) and (53), we obtain (49).

(50) follows from Lemma 3.1 and density of C∞(B+r ) in H1(B+

r ; t1−2s).

3.2. On the Almgren type frequency N . In this section, we shall study the differentiabilityof N , it’s limit at 0 and provide estimates of N ′.

Lemma 3.8. H ∈ C1(0, R) and

H ′(r) =2

rN+1−2s

∫

S+r

t1−2sw∂w

∂νdS, for every r ∈ (0, R),(54)

H ′(r) =2

rD(r), for every r ∈ (0, R).(55)


Proof. Fix r0 ∈ (0, R) and consider the limit

(56) limr→r0

H(r)−H(r0)

r − r0= limr→r0

∫

SN+

θ1−2s1

|w(rθ)|2 − |w(r0θ)|2r − r0

dS.

Since w ∈ C1(RN+1+ ), then, for every θ ∈ SN+ ,

(57) limr→r0

|w(rθ)|2 − |w(r0θ)|2r − r0

= 2∂w

∂ν(r0θ)w(r0θ).

On the other hand, for any r ∈ (r0/2, R) and θ ∈ SN+ we have∣∣∣∣|w(rθ)|2 − |w(r0θ)|2

r − r0

∣∣∣∣ 6 2 supB+

R\B+r02

|w| · supB+

R\B+r02

∣∣∣∣∂w

∂ν

∣∣∣∣ 6 2 supB+

R\B+r02

|w| · supB+

R\B+r02

(∣∣∣∣t

|z|wt∣∣∣∣+∣∣∣∣∇xw · x

|z|

∣∣∣∣)

and hence, by (56), (57), Lemma 3.3 and the Dominated Convergence Theorem, we obtain that

H ′(r0) = 2

∫

SN+

θ1−2s1

∂w

∂ν(r0θ)w(r0θ)dS(θ) =

2

rN+1−2s0

∫

S+r0

t1−2sw∂w

∂νdS.

The continuity of H ′ on the interval (0, R) follows by the representation of H ′ given above, Lemma3.3, and the Dominated Convergence Theorem.

Finally, (55) follows from (54), (39), and (50).

The regularity of the function D is established in the following lemma.

Lemma 3.9. The function D defined in (39) belongs to W 1,1loc(0, R) and

D′(r) =2

rN+1−2s

[r

∫

S+r

t1−2s

∣∣∣∣∂w

∂ν

∣∣∣∣2

dS − κs

∫

B′

r

(sh+

1

2(∇h · x)

)w2 dx

](58)

+κs

rN+1−2s

∫

B′

r

((N − 2s)f(x,w)w − 2NF (x,w)− 2∇xF (x,w) · x

)dx

+κs

rN−2s

∫

∂B′

r

(2F (x,w) − f(x,w)w

)dS′

in a distributional sense and for a.e. r ∈ (0, R).

Proof. For any r ∈ (0, r0) let

I(r) =

∫

B+r


∫

B′

r

(λ

|x|2sw2 + hw2 + f(x,w)w

)dx.(59)

From the fact that w ∈ H1(B+R ; t

1−2s), Lemma 2.5, and (8), we deduce that I ∈W 1,1(0, R) and

(60) I ′(r) =

∫

S+r

t1−2s|∇w|2 dS − κs

∫

∂B′

r

(λ

|x|2sw2 + hw2 + f(x,w)w

)dS′

for a.e. r ∈ (0, R) and in the distributional sense. Therefore D ∈W 1,1loc (0, R) and, using (49), (59),

and (60) into

D′(r) = r2s−1−N [−(N − 2s)I(r) + rI ′(r)],

we obtain (58) for a.e. r ∈ (0, R) and in the distributional sense.

Before going on, we recall that w is nontrivial and satisfies (38). We prove now that, if w 6≡ 0,H(r) does not vanish for r sufficiently small.

Lemma 3.10. There exists R0 ∈ (0, R) such that H(r) > 0 for any r ∈ (0, R0), where H is definedby (40).

Proof. Clearly from assumption (2), there exists R0 ∈ (0, R) such that

λ

ΛN,s+ChR

ε0

ΛN,s+ CfS

−1N,s

(ωN−1

N

)2∗(s)−p2∗(s) R

N(2∗(s)−p)2∗(s)

0 ‖w‖p−2

L2∗(s)(B′

R0)< 1,(61)

where ωN−1 denotes the volume of the unit sphere SN−1, i.e. ωN−1 =

∫SN−1 dS.


Next suppose by contradiction that there exists r0 ∈ (0, R0) such that H(r0) = 0. Then w = 0a.e. on S+

r0 . From (50) it follows that∫

B+r0


∫

B′

r0

w2

|x|2s dx− κs

∫

B′

r0

(h(x)w2 + f(x,w)w

)dx = 0.

From Lemma 2.5, assumptions (3)-(4), Holder’s inequality, and (8), it follows that

0 =

∫

B+r0


∫

B′

r0

w2

|x|2s dx− κs

∫

B′

r0

(h(x)w2 + f(x,w)w

)dx

>

[1− λ

ΛN,s− ChR

ε0

ΛN,s− CfS

−1N,s

(ωN−1

N

)2∗(s)−p2∗(s) R

N(2∗(s)−p)2∗(s)

0 ‖w‖p−2

L2∗(s)(B′

R0)

] ∫

B+r0

t1−2s|∇w|2dz,

which, together with (61), implies w ≡ 0 in B+r0 by Lemma 2.4. Classical unique continuation

principles for second order elliptic equations with locally bounded coefficients (see e.g. [26]) allowto conclude that w = 0 a.e. in B+

R , a contradiction.

Letting R0 be as in Lemma 3.10 and recalling (41), the Almgren type frequency function

(62) N (r) =D(r)

H(r)

is well defined in (0, R0). Using Lemmas 3.8 and 3.9, we can now compute the derivative of N .

Lemma 3.11. The function N defined in (62) belongs to W 1,1loc (0, R0) and

N ′(r) = ν1(r) + ν2(r)(63)

in a distributional sense and for a.e. r ∈ (0, R0), where

ν1(r) =2r[ (∫

S+rt1−2s

∣∣∂w∂ν

∣∣2 dS)·(∫

S+rt1−2sw2 dS

)−(∫

S+rt1−2sw ∂w

∂ν dS)2 ]

(∫S+rt1−2sw2 dS

)2 ,(64)

ν1 > 0, and

ν2(r) =− κs

∫B′

r(2sh+∇h · x)|w|2 dx∫S+rt1−2sw2 dS

+ κsr∫∂B′

r

(2F (x,w) − f(x,w)w

)dS′

∫S+rt1−2sw2 dS

(65)

+ κs

∫B′

r

((N − 2s)f(x,w)w − 2NF (x,w) − 2∇xF (x,w) · x

)dx

∫S+rt1−2sw2 dS

.

Proof. From Lemmas 3.8, 3.10, and 3.9, it follows that N ∈ W 1,1loc (0, R0). From (55) it follows

that

N ′(r) =D′(r)H(r) −D(r)H ′(r)

(H(r))2=D′(r)H(r) − 1

2r(H′(r))2

(H(r))2

and the proof of the lemma easily follows from (54) and (58). Now it is easy to see that ν1 > 0 bySchwarz’s inequality.

We now prove that N (r) admits a finite limit as r → 0+. To this aim, the following estimate playsa crucial role.

Lemma 3.12. Let N be the function defined in (62). There exist R ∈ (0, R0) and a constantC > 0 such that

(66)

∫

B+r


∫

B′

r

(λ

|x|2sw2 + hw2 + f(x,w)w

)dx

> −(N − 2s

2r

)∫

S+r

t1−2sw2dS + C

(∫

B′

r

w2

|x|2s dx+

(∫

B′

r

|w|2∗(s) dx) 2

2∗(s)

),


(67)

∫

B+r


∫

B′

r

(λ

|x|2sw2 + hw2 + f(x,w)w

)dx

> −(N − 2s

2r

)∫

S+r

t1−2sw2dS + C

∫

B+r

t1−2s|∇w|2 dt dx,

and

(68) N (r) > −N − 2s

2

for every r ∈ (0, R).

Proof. From Corollary 2.7, (3), and (4), it follows that

∫

B+r


∫

B′

r

(λ

|x|2sw2 + hw2 + f(x,w)w

)dx+

(N − 2s

2r

)∫

S+r

t1−2sw2dS

>

(κs(ΛN,s − λ)

2− Chκsr

ε

)∫

B′

r

w2

|x|2s dx

+

(ΛN,s − λ

2(1 + ΛN,s)SN,s− Cfκs|B′

r|2∗(s)−p2∗(s) ‖w‖p−2

L2∗(s)(B′

r)

)(∫

B′

r

|w|2∗(s) dx) 2

2∗(s)

for every r ∈ (0, R0). Since λ < ΛN,s, from the above estimate it follows that we can choose

R ∈ (0, R0) sufficiently small such that estimate (66) holds for r ∈ (0, R) for some positive constantC > 0. The proof of (67) can be performed in a similar way, using Lemmas 2.5 and 2.6. Estimate(66), together with (39) and (40), yields (68).

Lemma 3.13. Let R be as in Lemma 3.12 and ν2 as in (65). Then there exist a positive constant

C1 > 0 and a function g ∈ L1(0, R), g > 0 a.e. in (0, R), such that

|ν2(r)| 6 C1

[N (r) +

N − 2s

2

](r−1+ε + r

−1+2s(p0−2∗(s))

p0 + g(r))

for a.e. r ∈ (0, R) and

∫ r

0

g(ρ) dρ 61

1− α‖w‖p(1−α)Lp(B′

R) rN(

α2∗(s)−22∗(s)

−pα−2p0

)

for all r ∈ (0, R) with some α satisfying 2p < α < 1.

Proof. From (3) and (66) we deduce that

∣∣∣∣∣

∫

B′

r

(2sh(x) +∇h(x) · x)|w|2 dx∣∣∣∣∣ 6 2Chr

ε

∫

B′

r

|w|2|x|2s dx

6 2ChC−1rε+N−2s

[D(r) + N−2s

2 H(r)],

and, therefore, for any r ∈ (0, R), we have that

∣∣∣∣∣

∫B′

r(2sh(x) +∇h(x) · x)|w|2 dx

∫S+rt1−2sw2 dS

∣∣∣∣∣ 6 2ChC−1r−1+εD(r) + N−2s

2 H(r)

H(r)(69)

= 2ChC−1r−1+ε

[N (r) +

N − 2s

2

].


By (4), Holder’s inequality, and (66), for some constant const = const (N, s, Cf ) > 0 depending

on N, s, Cf , and for all r ∈ (0, R), there holds∣∣∣∣∫

B′

r


)dx

∣∣∣∣

6 const

∫

B′

r

(|w|2 + |w|2∗(s)) dx

6 const

((ωN−1

N

)2sN

r2s + ‖w‖2∗(s)−2

L2∗(s)(B′

R)

)(∫

B′

r

|w|2∗(s)dx) 2

2∗(s)

6const

C

((ωN−1

N

)2sN

r2s +(ωN−1

N

)2s(p0−2∗(s))Np0

r2s(p0−2∗(s))

p0 ‖w‖2∗(s)−2Lp0(B′

R)

)rN−2s

[D(r) + N−2s

2 H(r)]

and hence

(70)

∣∣∣∣∣

∫B′

r


)dx

∫S+rt1−2sw2 dS

∣∣∣∣∣

6const

C

((ωN−1

N

)2sN

r2s2∗(s)

p0 +(ωN−1

N

)2s(p0−2∗(s))Np0 ‖w‖2

∗(s)−2Lp0(B′

R)

)r−1+

2s(p0−2∗(s))

p0

[N (r) + N−2s

2

].

Let us fix 2p < α < 1. By Holder’s inequality and (66),

(∫

B′

r

|w|p dx)α

=

(∫

B′

r

|w|p− 2α |w| 2

α dx

)α(71)

6

(∫

B′

r

|w|2∗(s) pα−2

2∗(s)α−2 dx

)α2∗(s)−22∗(s)

(∫

B′

r

|w|2∗(s) dx) 2

2∗(s)

6

(ωN−1

N

)α2∗(s)−22∗(s)

−pα−2p0

rN(

α2∗(s)−22∗(s)

− pα−2p0

)‖w‖pα−2

Lp0(B′

R)

rN−2s

C

[D(r) + N−2s

2 H(r)]

= C−1(ωN−1

N

)βN

r−1+β

[N (r) +

N − 2s

2

](∫

S+r

t1−2sw2 dS

)

for all r ∈ (0, R), where β = N(α2∗(s)−2

2∗(s) − pα−2p0

)> 0. From (4), (71), and (68), there exists some

const = const (N, s, Cf ) > 0 depending on N, s, Cf such that, for all r ∈ (0, R),

(72)

∣∣∣∣∣r∫∂B′

r

(2F (x,w) − f(x,w)w

)dS′

∫S+rt1−2sw2 dS

∣∣∣∣∣ 6 constr∫∂B′

r|w|p dS′

∫S+rt1−2sw2 dS

6const

C

(ωN−1

N

)βN

[N (r) +

N − 2s

2

] rβ∫∂B′

r|w|p dS′

( ∫B′

r|w|p dx

)α .

By a direct calculation, we have that

(73)rβ∫∂B′

r|w|p dS′

( ∫B′

r|w|p dx

)α =1

1− α

[d

dr

(rβ(∫

B′

r

|w|p dx)1−α)

− β r−1+β

(∫

B′

r

|w|p dx)1−α]

in the distributional sense and for a.e. r ∈ (0, R). Since

limr→0+

rβ(∫

B′

r

|w|p dx)1−α

= 0

we deduce that the function

r 7→ d

dr

(rβ(∫

B′

r

|w|p dx)1−α)


is integrable over (0, R). Being

r−1+β

(∫

B′

r

|w|p dx)1−α

= o(r−1+β)

as r → 0+, we have that also the function

r 7→ r−1+β

(∫

B′

r

|w|p dx)p−2

p

is integrable over (0, R). Therefore, by (73), we deduce that

(74) g(r) :=rβ∫∂B′

r|w|p dS′

( ∫B′

r|w|p dx

)α ∈ L1(0, R)

and

0 6

∫ r

0

g(ρ) dρ 61

1− α‖w‖p(1−α)Lp(B′

r)rβ(75)

for all r ∈ (0, R). Collecting (69), (70), (72), (74), and (75), we obtain the stated estimate.

Lemma 3.14. Let R be as in Lemma 3.12 and N as in (62). Then there exist a positive constantC2 > 0 such that

(76) N (r) 6 C2

for all r ∈ (0, R).

Proof. By Lemma 3.11 and Lemma 3.13, we obtain

(77)

(N +

N − 2s

2

)′(r) > ν2(r) > −C1

[N (r) +

N − 2s

2

](r−1+ε + r

−1+2s(p0−2∗(s))

p0 + g(r))

for a.e. r ∈ (0, R). Integration over (r, R) yields

N (r) 6 −N − 2s

2+

(N (R) +

N − 2s

2

)exp

C1

(Rε

ε+p0R

2s(p0−2∗(s))p0

2s(p0 − 2∗(s))+

∫ R

0

g(ρ) dρ

)

for any r ∈ (0, R), thus proving estimate (76).

Lemma 3.15. The limit

γ := limr→0+

N (r)

exists and is finite.

Proof. By Lemmas 3.13 and 3.14, the function ν2 defined in (65) belongs to L1(0, R). Hence,

by Lemma 3.11, N ′ is the sum of a nonnegative function and of a L1-function on (0, R). Therefore

N (r) = N (R)−∫ R

r

N ′(ρ) dρ

admits a limit as r → 0+ which is necessarily finite in view of (68) and (76).

The function H defined in (40) can be estimated as follows.

Lemma 3.16. Let γ := limr→0+ N (r) be as in Lemma 3.15 and R as in Lemma 3.12. Then thereexists a constant K1 > 0 such that

(78) H(r) 6 K1r2γ for all r ∈ (0, R).

Moreover, for any σ > 0 there exists a constant K2(σ) > 0 depending on σ such that

(79) H(r) > K2(σ) r2γ+σ for all r ∈ (0, R).


Proof. By Lemma 3.15, N ′ ∈ L1(0, r) and, by Lemma 3.14, N is bounded, then from (77) and(75) it follows that

(80) N (r) − γ =

∫ r

0

N ′(ρ) dρ > −C3rδ

for some constant C3 > 0 and all r ∈ (0, R), where

(81) δ = min

ε,

2s(p0 − 2∗(s))

p0, N

(α2∗(s)− 2

2∗(s)− pα− 2

p0

)> 0.

Therefore by (55), (62), and (80) we deduce that, for all r ∈ (0, R),

H ′(r)

H(r)=

2N (r)

r>

2γ

r− 2C3r

−1+δ,

which, after integration over the interval (r, R), yields (78).Since γ = limr→0+ N (r), for any σ > 0 there exists rσ > 0 such that N (r) < γ + σ/2 for any

r ∈ (0, rσ) and henceH ′(r)

H(r)=

2N (r)

r<

2γ + σ

rfor all r ∈ (0, rσ).

Integrating over the interval (r, rσ) and by continuity of H outside 0, we obtain (79) for someconstant K2(σ) depending on σ.

4. The blow-up argument

The main result of this section, which also contains Theorem 1.1, is the following theorem.

Theorem 4.1. Let w satisfy (38), with s, λ, h, f as in assumptions (2), (3), and (4). Then, lettingN (r) as in (41), there there exists k0 ∈ N, k0 > 1, such that

(82) limr→0+

N (r) = −N − 2s

2+

√(N − 2s

2

)2+ µk0(λ).

Furthermore, if γ denotes the limit in (82), m > 1 is the multiplicity of the eigenvalue µj0(λ) =

µj0+1(λ) = · · · = µj0+m−1(λ) and ψij0+m−1i=j0

(j0 6 k0 6 j0 + m − 1) is an L2(SN+ ; θ1−2s1 )-

orthonormal basis for the eigenspace of problem (10) associated to µk0(λ), then

τ−γw(0, τx) → |x|γj0+m−1∑

i=j0

βiψi

(0,

x

|x|)

in C1,αloc (B

′1 \ 0) as τ → 0+,

τ−γw(τθ) →j0+m−1∑

i=j0

βiψi(θ) in C0,α(SN+ ) as τ → 0+,

τ−γw(0, τθ′) →j0+m−1∑

i=j0

βiψi(0, θ′) in C1,α(SN−1) as τ → 0+,

and

τ1−γ∇xw(0, τθ′) →

j0+m−1∑

i=j0

βi

(γψi(0, θ

′)θ′ +∇SN−1ψi(0, ·)(θ′))

in C0,α(SN−1) as τ → 0+,

for some α ∈ (0, 1), where

βi = R−γ∫

SN+

θ1−2s1 w(R θ)ψi(θ) dS(θ)

+ κs

∫

SN−1

[∫ R

0

h(tθ′)w(0, tθ′) + f(tθ′, w(0, tθ′))

2γ +N − 2s

(t2s−γ−1 − tγ+N−1

R2γ+N−2s

)dt

]ψi(0, θ

′) dS(θ′),

for all R > 0 such that B′R = x ∈ R

N : |x| 6 R ⊂ Ω and (βj0 , βj0+1, . . . , βj0+m−1) 6= (0, 0, . . . , 0).


To prove Theorem 4.1, we start by determining the asymptotic profile of blowing up renormalizedsolutions to (37).

Lemma 4.2. Let w as in Theorem 4.1. Let γ := limr→0+ N (r) as in Lemma 3.15. Then

(i) there exists k0 ∈ N, k0 > 1, such that γ = −N−2s2 +

√(N−2s

2

)2+ µk0(λ);

(ii) for every sequence τn → 0+, there exist a subsequence τnkk∈N and an eigenfunction ψ of

problem (10) associated to the eigenvalue µk0(λ) such that ‖ψ‖L2(SN+ ;θ1−2s1 ) = 1 and

w(τnkz)√

H(τnk)→ |z|γψ

( z|z|)

strongly in H1(B+r ; t

1−2s) and in C0,αloc (B

+r \ 0) for some α ∈ (0, 1) and all r ∈ (0, 1) and

w(0, τnkx)√

H(τnk)→ |x|γψ

(0,

x

|x|)

in C1,αloc (B

′1 \ 0).

Proof. Let us set

(83) wτ (z) =w(τz)√H(τ)

.

We notice that∫S+1t1−2s|wτ |2dS = 1. Moreover, by scaling and (76),

(84)

∫

B+1

t1−2s|∇wτ (z)|2dz − κs

∫

B′

1

(λ

|x|2s |wτ |2 + τ2sh(τx)|wτ |2

+τ2s√H(τ)

f(τx,√H(τ)wτ

)wτ)dx = N (τ) 6 C2

for every τ ∈ (0, R), whereas, from (67),

(85) N (τ) >τ−N+2s

H(τ)

(−(N − 2s

2τ

)∫

S+τ

t1−2sw2dS + C

∫

B+τ

t1−2s|∇w|2 dt dx)

= −N − 2s

2+ C

∫

B+1

t1−2s|∇wτ (z)|2dz

for every τ ∈ (0, R). From (84) and (85) we deduce that

(86) wττ∈(0,R) is bounded in H1(B+1 ; t

1−2s).

Therefore, for any given sequence τn → 0+, there exists a subsequence τnk→ 0+ such that

wτnk w weakly in H1(B+1 ; t

1−2s) for some w ∈ H1(B+1 ; t1−2s). Due to compactness of the trace

embedding (28), we obtain that∫S+1t1−2s|w|2dS = 1. In particular w 6≡ 0.

For every small τ ∈ (0, R), wτ satisfies

(87)

div(t1−2s∇wτ ) = 0, in B+

1 ,

− limt→0+ t1−2s ∂wτ

∂t = κs

(λ

|x|2swτ + τ2sh(τx)wτ + τ2s√

H(τ)f(τx,

√H(τ)wτ )

), on B′

1,

in a weak sense, i.e.

(88)

∫

B+1

t1−2s∇wτ · ∇ϕ dt dx

= κs

∫

B′

1

(λ

|x|2swτ + τ2sh(τx)wτ +

τ2s√H(τ)

f(τx,√H(τ)wτ

))ϕ(0, x) dx


for all ϕ ∈ H1(B+1 ; t

1−2s) s.t. ϕ = 0 on SN+ and, for such ϕ, by (4) and Holder’s inequality,

(89)τ2s√H(τ)

∣∣∣∣∫

B′

1

f(τx,√H(τ)wτ (0, x)

)ϕ(0, x) dx

∣∣∣∣

6 Cfτ2s

∫

B′

1

|w(0, τx)|p−2|wτ (0, x)||ϕ(0, x)| dx

6 Cf‖Tr ϕ‖Lp(B′

1)‖wτ‖Lp(B′

1)‖w‖p−2

Lp(B′

τ )τ

(N−2s)(2∗(s)−p)p = o(1) as τ → 0+

and, by (3) and Lemma 2.5,

(90) τ2s

∣∣∣∣∣

∫

B′

1

h(τx)wτ ϕ(0, x) dx

∣∣∣∣∣

6Chτ

ε

κsΛN,s

(∫

B+1

t1−2s|∇wτ (z)|2dz + N − 2s

2

)1/2(∫

B+1

t1−2s|∇ϕ(z)|2dz)1/2

= o(1) as τ → 0+.

From (89), (90), and weak convergence wτnk w in H1(B+1 ; t

1−2s), we can pass to the limit in(87) along the sequence τnk

and obtain that w weakly solves

(91)

div(t1−2s∇w) = 0, in B+

1 ,

− limt→0+ t1−2s ∂w

∂t = κsλ

|x|2s w, on B′1.

From (4), letting q = p0p−2 >

N2s with p0 as in Lemma 3.5, we have that

∥∥∥∥∥τ2s√H(τ)

f(τx,√H(τ)wτ (x)

)

wτ (x)

∥∥∥∥∥Lq(B′

2)

6 Cfτ2s−N

q

(∫

B2τ

|w(x)|p0dx)1/q)

= O(1)

as τ → 0+. Therefore from Lemma 3.3 part (i) there holds

(92) wτnk → w in C0,αloc (B

+r \ 0),

while Lemma 3.3 part (ii) and Remark 3.4 imply that

(93) ∇xwτnk → ∇xw, and t1−2s ∂w

τnk

∂t→ t1−2s ∂w

∂tin C0,α

loc (B+r \ 0)

for some α ∈ (0, 1) and all r ∈ (0, 1). Reasoning as in (89), (90), we can prove that

(94)τ2s√H(τ)

∫

B′

1

f(τx,√H(τ)wτ

)wτ dx = o(1) as τ → 0+

and

(95) τ2s∫

B′

1

h(τx)|wτ |2 dx = o(1) as τ → 0+.

Multiplying equation (87) with wτ , integrating in B+r , and using (93), (94), (95), we easily obtain

that ‖wτnk ‖H1(B+r ;t1−2s) → ‖w‖H1(B+

r ;t1−2s) for all r ∈ (0, 1), and hence

(96) wτnk → w in H1(B+r ; t

1−2s)

for any r ∈ (0, 1).For any r ∈ (0, 1) and k ∈ N, let us define the functions

Dk(r) =1

rN−2s

[∫

B+r

t1−2s|∇wτnk |2 dt dx− κs

∫

B′

r

(λ

|x|2s |wτnk |2 + τ2snk

h(τnkx)|wτnk |2

+τ2snk√H(τnk

)f(τnk

x,√H(τnk

)wτnk

)wτnk

)dx

]

and

Hk(r) =1

rN+1−2s

∫

S+r

t1−2s|wτnk |2 dS.


Direct calculations yield

(97) Nk(r) :=Dk(r)

Hk(r)=D(τnk

r)

H(τnkr)

= N (τnkr) for all r ∈ (0, 1).

From (96), (94), and (95), it follows that, for any fixed r ∈ (0, 1),

(98) Dk(r) → D(r),

where

(99) D(r) =1

rN−2s

[ ∫

B+r


∫

B′

r

λ

|x|2s w2 dx

]for all r ∈ (0, 1).

On the other hand, by compactness of the trace embedding (28), we also have

(100) Hk(r) → H(r) for any fixed r ∈ (0, 1),

where

(101) H(r) =1

rN+1−2s

∫

S+r

t1−2sw2 dS.

From (35) it follows that D(r) > −N−2s2 H(r) for all r ∈ (0, 1). Therefore, if, for some r ∈ (0, 1),

H(r) = 0 then D(r) > 0, and passing to the limit in (97) should give a contradiction with Lemma3.15. Hence Hw(r) > 0 for all r ∈ (0, 1) and the function

N (r) :=D(r)

H(r)

is well defined for r ∈ (0, 1). From (97), (98), (100), and Lemma 3.15, we deduce that

(102) N (r) = limk→∞

N (τnkr) = γ

for all r ∈ (0, 1). Therefore N is constant in (0, 1) and hence N ′(r) = 0 for any r ∈ (0, 1). By (91)and Lemma 3.11 with h ≡ 0 and f ≡ 0, we obtain

(∫

S+r

t1−2s

∣∣∣∣∂w

∂ν

∣∣∣∣2

dS

)·(∫

S+r

t1−2sw2 dS

)−(∫

S+r

t1−2sw∂w

∂νdS

)2= 0 for all r ∈ (0, 1),

which implies that w and ∂w∂ν have the same direction as vectors in L2(S+

r ; t1−2s) and hence there

exists a function η = η(r) such that ∂w∂ν (r, θ) = η(r)w(r, θ) for all r ∈ (0, 1) and θ ∈ SN+ . After

integration we obtain

(103) w(r, θ) = e∫

r1η(s)dsw(1, θ) = ϕ(r)ψ(θ), r ∈ (0, 1), θ ∈ S

N+ ,

where ϕ(r) = e∫

r1η(s)ds and ψ(θ) = w(1, θ). From (91), (103), and Lemma 2.1, it follows that,

weakly, 1rN

(rN+1−2sϕ′)′θ1−2s

1 ψ(θ) + r−1−2sϕ(r) divSN (θ1−2s1 ∇SNψ(θ)) = 0,

− limθ1→0+ θ1−2s1 ∇SNψ(θ) · e1 = κsλψ(0, θ

′).

Taking r fixed we deduce that ψ is an eigenfunction of the eigenvalue problem (10). If µk0(λ) isthe corresponding eigenvalue then ϕ(r) solves the equation

1

rN(rN+1−2sϕ′)′ − µk0(λ)r

−1−2sϕ(r) = 0

i.e.

ϕ′′(r) +N + 1− 2s

rϕ′ − µk0(λ)

r2ϕ(r) = 0

and hence ϕ(r) is of the form

ϕ(r) = c1rσ+k0 + c2r

σ−

k0

for some c1, c2 ∈ R, where

σ+k0

= −N − 2s

2+

√(N − 2s

2

)2+ µk0(λ) and σ−

k0= −N − 2s

2−√(

N − 2s

2

)2+ µk0(λ).


Since the function |x|σ−

k0ψ( x|x|) /∈ L2(B′1; |x|−2s) and hence |z|σ

−

k0ψ( z|z| ) /∈ H1(B+1 ; t1−2s) in virtue

of Lemma 2.5, we deduce that c2 = 0 and ϕ(r) = c1rσ+k0 . Moreover, from ϕ(1) = 1, we obtain that

c1 = 1 and then

(104) w(r, θ) = rσ+k0ψ(θ), for all r ∈ (0, 1) and θ ∈ S

N+ .

It remains to prove part (i). From (104) and the fact that∫SN+θ1−2s1 ψ2(θ)dS = 1 it follows that

D(r) =1

rN−2s

[ ∫

B+r


∫

B′

r

λ

|x|2s w2 dx

]

= r2s−N (σ+k0)2∫ r

0

tN−1−2s+2σ+

k0 dt

+ r2s−N(∫ r

0

tN−1−2s+2σ+

k0dt

)(∫

SN+

θ1−2s1 |∇SNψ(θ)|2 dS − λκs

∫

SN−1

|T ψ(θ′)|2 dS′)

=(σ+k0)2 + µk0(λ)

N − 2s+ 2σ+k0

r2σ+

k0 = σ+k0r2σ+

k0

and

H(r) =

∫

SN+

θ1−2s1 w2(rθ) dS = r

2σ+k0 ,

and hence from (102) it follows that γ = N (r) = D(r)

H(r)= σ+

k0. This completes the proof of the

lemma.

The following lemma describes the behavior of H(r) as r → 0+.

Lemma 4.3. Let w satisfy (37), H be defined in (40), and let γ := limr→0+ N (r) as in Lemma3.15. Then the limit

limr→0+

r−2γH(r)

exists and it is finite.

Proof. In view of (78) it is sufficient to prove that the limit exists. By (40), (55), and Lemma 3.15we have

d

dr

H(r)

r2γ= −2γr−2γ−1H(r) + r−2γH ′(r) = 2r−2γ−1(D(r) − γH(r)) = 2r−2γ−1H(r)

∫ r

0

N ′(ρ)dρ.

Integration over (r, R) yields

(105)H(R)

R2γ− H(r)

r2γ=

∫ R

r

2s−2γ−1H(ρ)

(∫ ρ

0

ν1(t)dt

)dρ+

∫ R

r

2ρ−2γ−1H(ρ)

(∫ ρ

0

ν2(t)dt

)dρ

where ν1 and ν2 are as in (64) and (65). Since, by Schwarz’s inequality, ν1 > 0, we have that

limr→0+∫ Rr

2ρ−2γ−1H(ρ)(∫ ρ

0ν1(t)dt

)dρ exists. On the other hand, by (78), Lemma 3.13, and

(76), we deduce that∣∣∣∣ρ

−2γ−1H(ρ)

(∫ ρ

0

ν2(t)dt

)∣∣∣∣ 6 K1C1

(C2 +

N − 2s

2

)ρ−1

∫ ρ

0

(t−1+ε + t

−1+2s(p0−2∗(s))

p0 + g(t))dt

6 K1C1

(C2 +

N − 2s

2

)ρ−1

(ρε

ε+p0ρ

2s(p0−2∗(s))p0

2s(p0 − 2∗(s))+

1

1− α‖w‖p(1−α)Lp(B′

R) ρN(

α2∗(s)−22∗(s)

− pα−2p0

))

for all ρ ∈ (0, R), which proves that ρ−2γ−1H(ρ)(∫ ρ

0 ν2(t)dt)∈ L1(0, R). Hence both terms at the

right hand side of (105) admit a limit as r → 0+ thus completing the proof.

From Lemma 4.2, the following pointwise estimate for solutions to (1) and (38) can be derived.

Lemma 4.4. Let w satisfying (38). Then there exist C4, C5 > 0 and r ∈ (0, R) such that

(i) supS+r|w|2 6 C4

rN+1−2s

∫S+rt1−2s|w(z)|2 dS for every 0 < r < r,


(ii) |w(z)| 6 C5|z|γ for all z ∈ B+r and in particular |w(0, x)| 6 C5|x|γ for all x ∈ B′

r, whereγ := limr→0+ N (r) is as in Lemma 3.15.

Proof. We first notice that (ii) follows directly from (i) and (78). In order to prove (i), we argueby contradiction and assume that there exists a sequence τn → 0+ such that

supθ∈SN+

∣∣∣w( τn2θ)∣∣∣

2

> nH(τn2

)

with H as in (40), i.e.

supx∈S+

1/2

|w(τnz)|2 > 2N+1−2sn

∫

S+1/2

t1−2sw2(τnz)dS,

i.e., defining wτ as in (83)

(106) supx∈S+

1/2

|wτn(z)|2 > 2N+1−2sn

∫

S+1/2

t1−2s|wτn(z)|2dS.

From Lemma 4.2, along a subsequence τnkwe have that wτnk → |z|γψ

(z|z|)in C0,α

loc (S+1/2), for some

ψ eigenfunction of problem (10), hence passing to the limit in (106) gives rise to a contradiction.

We will now prove that limr→0+ r−2γH(r) is strictly positive.

Lemma 4.5. Under the same assumption as in Lemmas 4.3 and 4.4, we have

limr→0+

r−2γH(r) > 0.

Proof. For all k > 1, let ψk be as in (24), i.e. ψk is a L2(SN+ ; θ1−2s1 )-normalized eigenfunction of

problem (10) associated to the eigenvalue µk(λ) and ψkk is an orthonormal basis of L2(SN+ ; θ1−2s1 ).

From Lemma 4.2 there exist j0,m ∈ N, j0,m > 1 such that m is the multiplicity of the eigenvalueµj0(λ) = µj0+1(λ) = · · · = µj0+m−1(λ) and

(107) γ = limr→0+

N (r) = −N − 2s

2+

√(N − 2s

2

)2+ µi(λ), i = j0, . . . , j0 +m− 1.

Let us expand w as

w(z) = w(τθ) =

∞∑

k=1

ϕk(τ)ψk(θ)

where τ = |z| ∈ (0, R], θ = z/|z| ∈ SN+ , and

(108) ϕk(τ) =

∫

SN+

θ1−2s1 w(τ θ)ψk(θ) dS.

The Parseval identity yields

(109) H(τ) =

∫

SN+

θ1−2s1 w2(τθ) dS =

∞∑

k=1

ϕ2k(τ), for all 0 < τ 6 R.

In particular, from (78) and (109) it follows that, for all k > 1,

(110) ϕk(τ) = O(τγ ) as τ → 0+.

Equations (38) and (24) imply that, for every k,

−ϕ′′k(τ) −

N + 1− 2s

τϕ′k(τ) +

µk(λ)

τ2ϕk(τ) = ζk(τ), in (0, R),

where

(111) ζk(τ) =κs

τ2−2s

∫

SN−1

(h(τθ′)w(0, τθ′) + f(τθ′, u(τθ′))

)ψk(0, θ

′) dS′.

A direct calculation shows that, for some ck1 , ck2 ∈ R,

(112) ϕk(τ) = τσ+k

(ck1 +

∫ R

τ

t−σ+k +1

σ+k − σ−

k

ζk(t) dt

)+ τσ

−

k

(ck2 +

∫ R

τ

t−σ−

k +1

σ−k − σ+

k

ζk(t) dt

),


where

(113) σ+k = −N − 2s

2+

√(N − 2s

2

)2+ µk(λ) and σ−

k = −N − 2s

2−√(

N − 2s

2

)2+ µk(λ).

From (3), (4), Lemma 4.4, (107), and the fact that, in view of (23),

2s+ (p− 2)γ =(N − 2s)(2∗(s)− p)

2+ (p− 2)

√(N − 2s

2

)2+ µj0(λ) > 0,

we deduce that, for all i = j0, . . . , j0 +m− 1,

(114) ζi(τ) = O(τ−2+δ+σ+i ) as τ → 0+,

with δ = minε, 2s+ (p− 2)γ > 0. Consequently, the functions

t 7→ t−σ+i +1

σ+i − σ−

i

ζi(t) and t 7→ t−σ−

i +1

σ−i − σ+

i

ζi(t)

belong to L1(0, R). Hence

τσ+i

(ci1 +

∫ R

τ

ρ−σ+i +1

σ+i − σ−

i

ζi(ρ) dρ

)= o(τσ

−

i ) as τ → 0+,

and then, by (110), there must be

ci2 = −∫ R

0

t−σ−

i +1

σ−i − σ+

i

ζi(t) dt.

Using (114), we then deduce that

τσ−

i

(ci2 +

∫ R

τ

t−σ−

i +1

σ−i − σ+

i

ζi(t) dt

)= τσ

−

i

(∫ τ

0

t−σ−

i +1

σ+i − σ−

i

ζi(t) dt

)= O(τσ

+i +δ)(115)

as τ → 0+. From (112) and (115), we obtain that, for all i = j0, . . . , j0 +m− 1,

(116) ϕi(τ) = τσ+i

(ci1 +

∫ R

τ

t−σ+i +1

σ+i − σ−

i

ζi(t) dt+O(τ δ)

)as τ → 0+.

Let us assume by contradiction that limλ→0+ λ−2γH(λ) = 0. Then, for all i ∈ j0, . . . , j0+m− 1,

(107) and (109) would imply that

limτ→0+

τ−σ+i ϕi(τ) = 0.

Hence, in view of (116),

ci1 +

∫ R

0

t−σ+i +1

σ+i − σ−

i

ζi(t) dt = 0,

which, together with (114), implies

τσ+i

(ci1 +

∫ R

τ

t−σ+i +1

σ+i − σ−

i

ζi(t) dt

)= τσ

+i

∫ τ

0

t−σ+i +1

σ−i − σ+

i

ζi(t) dt = O(τσ+i +δ)(117)

as τ → 0+. Collecting (112), (115), and (117), we conclude that

ϕi(τ) = O(τσ+i +δ) as τ → 0+ for every i ∈ j0, . . . , j0 +m− 1,

namely, √H(τ) (wτ , ψ)L2(SN+ ;θ1−2s

1 ) = O(τγ+δ) as τ → 0+

for every ψ ∈ A0 = spanψij0+m−1i=j0

, where A0 is the eigenspace of problem (10) associated to

the eigenvalue µj0(λ) = µj0+1(λ) = · · · = µj0+m−1(λ). From (79), there exists C(δ) > 0 such that√H(τ) > C(δ)τγ+

δ2 for τ small, and therefore

(118) (wτ , ψ)L2(SN+ ;θ1−2s1 ) = O(τ

δ2 ) as τ → 0+


for every ψ ∈ A0. From Lemma 4.2, for every sequence τn → 0+, there exist a subsequence

τnkk∈N and an eigenfunction ψ ∈ A0

(119)

∫

SN+

θ1−2s1 ψ2(θ)dS = 1 and wτnk → ψ in L2(SN+ ; θ1−2s

1 ).

From (118) and (119), we infer that

0 = limk→+∞

(wτnk , ψ)L2(SN−1) = ‖ψ‖2L2(SN+ ;θ1−2s

1 )= 1,

thus reaching a contradiction.

We can now completely describe the behavior of solutions to (38) near the singularity, hence provingTheorem 4.1.

Proof of Theorem 4.1. Identity (82) follows from part (i) of Lemma 4.2, thus there exists

k0 ∈ N, k0 > 1, such that γ = limr→0+ N (r) = −N−2s2 +

√(N−2s

2

)2+ µk0(λ). Let us denote as

m the multiplicity of µj0(λ) so that, for some j0 ∈ N, j0 > 1, j0 6 k0 6 j0 + m − 1, µj0(λ) =

µj0+1(λ) = · · · = µj0+m−1(λ) and let ψij0+m−1i=j0

be an L2(SN+ ; θ1−2s1 )-orthonormal basis for the

eigenspace associated to µk0(λ).Let τnn∈N ⊂ (0,+∞) such that limn→+∞ τn = 0. Then, from part (ii) of Lemma 4.2 and

Lemmas 4.3 and 4.5, there exist a subsequence τnkk∈N and m real numbers βj0 , . . . , βj0+m−1 ∈ R

such that (βj0 , βj0+1, . . . , βj0+m−1) 6= (0, 0, . . . , 0) and

τ−γnkw(0, τnk

x) → |x|γj0+m−1∑

i=j0

βiψi

(0,

x

|x|)

in C1,αloc (B

′1 \ 0) as k → +∞,(120)

τ−γnkw(τnk

θ) →j0+m−1∑

i=j0

βiψi(θ) in C0,α(SN+ ) as k → +∞,(121)

τ−γnkw(0, τnk

θ′) →j0+m−1∑

i=j0

βiψi(0, θ′) in C1,α(SN−1) as k → +∞,(122)

and

(123) τ1−γnk∇xw(0, τnk

θ′) →j0+m−1∑

i=j0

βi(γψi(0, θ′)θ′ +∇SN−1ψi(0, θ

′)) in C0,α(SN−1) as k → +∞

for some α ∈ (0, 1).We now prove that the βi’s depend neither on the sequence τnn∈N nor on its subsequence

τnkk∈N.

Defining ϕi and ζi as in (108) and (111), from (121) it follows that, for any i = j0, . . . , j0+m−1,

(124) τ−γnkϕi(τnk

) =

∫

SN+

θ1−2s1

w(τnkθ)

τγnk

ψi(θ) dS →j0+m−1∑

j=j0

βj

∫

SN+

θ1−2s1 ψj(θ)ψi(θ) dS = βi

as k → +∞. As deduced in the proof of Lemma 4.5, for any i = j0, . . . , j0 +m− 1 and τ ∈ (0, R]there holds

ϕi(τ) = τσ+i

(ci1 +

∫ R

τ

t−σ+i +1

σ+i − σ−

i

ζi(t) dt

)+ τσ

−

i

(∫ τ

0

t−σ−

i +1

σ+i − σ−

i

ζi(t) dt

)(125)

= τσ+i

(ci1 +

∫ R

τ

t−σ+i +1

σ+i − σ−

i

ζi(t) dt+O(τ δ)

)as τ → 0+,

for some ci1 ∈ R, where σ±i are defined in (113). Choosing τ = R in the first line of (125), we

obtain

ci1 = R−σ+i ϕi(R)−Rσ

−

i −σ+i

∫ R

0

s−σ−

i +1

σ+i − σ−

i

ζi(s) ds.


Hence (125) yields

τ−γϕi(τ) → R−σ+i ϕi(R)−Rσ

−

i −σ+i

∫ R

0

t−σ−

i +1

σ+i − σ−

i

ζi(t) dt+

∫ R

0

t−σ+i +1

σ+i − σ−

i

ζi(t) dt as τ → 0+,

and therefore from (124) we deduce that

βi = R−γ∫

SN+

θ1−2s1 w(Rθ)ψi(θ) dS

−R−2γ−N+2s

∫ R

0

κsργ+N−1

2γ +N − 2s

(∫

SN−1

(h(ρθ′)w(0, ρθ′) + f(ρθ′, w(0, ρθ′))

)ψi(0, θ

′) dS′)dρ

+

∫ R

0

κsρ2s−γ−1

2γ +N − 2s

(∫

SN−1

(h(ρθ′)w(0, ρθ′) + f(ρθ′, w(0, ρθ′))

)ψi(0, θ

′) dS′)dρ.

In particular the βi’s depend neither on the sequence τnn∈N nor on its subsequence τnkk∈N,

thus implying that the convergences in (120), (121), (122), and (123) actually hold as τ → 0+ andproving the theorem.

5. Proof of Theorem 1.1

Let Ds,2(Ω) denote the completion on C∞c (Ω) with respect to the norm ‖ · ‖Ds,2(RN ). Simple

density arguments show that (7) is equivalent to

(126) (u, ϕ)Ds,2(RN ) =

∫

Ω

(λ

|x|2s u(x) + h(x)u(x) + f(x, u(x))

)ϕ(x) dx, for all ϕ ∈ Ds,2(Ω).

Since u ∈ Ds,2(RN ), we can let H(u) ∈ D1,2(RN+1+ ; t1−2s) such that

∫

RN+1+

t1−2s∇H(u) · ∇ϕdt dx = 0, for all ϕ ∈ C∞c (RN+1

+ ),

and H(u) = u on RN identified with ∂RN+1+ , i.e. H(u) weakly satisfies

div(t1−2s∇H(u)) = 0, in R

N+1+ ,

H(u) = u, on ∂RN+1+ = 0 × RN .

From [4] we have that∫

RN+1+

t1−2s∇H(u) · ∇ϕ dt dx = κs(u, ϕ)Ds,2(RN ) for all ϕ ∈ D1,2(RN+1+ ; t1−2s),

where

κs =Γ(1− s)

22s−1Γ(s),

i.e.

− limt→0+

t1−2s ∂H(u)

∂t= κs(−∆)su(x),

in a weak sense. Therefore u ∈ Ds,2(RN ) weakly solves (1) in Ω in the sense of (7) if and only ifits extension w = H(u) satisfies

(127)

div(t1−2s∇w) = 0, in RN+1+ ,

w = u, on RN ,

− limt→0+ t1−2s ∂w

∂t (t, x) = κs

(λ

|x|2sw + hw + f(x,w)), on Ω,

in a weak sense, i.e. if for all ϕ ∈ D1,2(RN+1+ ; t1−2s) such that x 7→ ϕ(0, x) ∈ Ds,2(Ω) we have

(128)

∫

RN+1+

t1−2s∇w · ∇ϕ dt dx = κs

∫

Ω

(λ


)ϕ dx.

Since D1,2(RN+1+ ; t1−2s) → H1(B+

R ; t1−2s) for all R > 0, the result follows from Theorem 4.1.


Proof of Theorem 1.2. The proof follows from the proof of Theorem 1.1, observing that, ifλ = 0, since no singularity occurs in the coefficients of the equation, the convergences in (92) and

(93) hold in C0,α(B+r ) by virtue of Lemma 3.3.

Proof of Theorem 1.3. The proof follows directly from Theorem 1.1. Indeed, let u be asolution to (1) satisfying u(x) = o(|x|n) = o(1)|x|n as |x| → 0 for all n ∈ N; if, by contradiction,u 6≡ 0 in Ω, convergence (13) in Theorem 1.1 would hold, thus contradicting the assumption that

u(x) = o(|x|n) if n > −N−2s2 +

√(2s−N

2

)2+ µk0(λ).

6. Proof of Theorem 1.4

Let u ∈ Ds,2(RN ) be a weak solution to (14) in Ω such that u ≡ 0 on a set E ⊂ Ω with L(E) > 0,where L denotes the N -dimensional Lebesgue measure. By Lebesgue’s density Theorem, for a.e.point x ∈ E there holds

limr→0+

L(E ∩B(x, r))

L(B(x, r))= 1 and lim

r→0+

L((RN \ E) ∩B(x, r))

L(B(x, r))= 0,

where B(x, r) = y ∈ RN : |y − x| < r, i.e. a.e. point of E is a density point of E. Let x0 be adensity point of E; then for all ε > 0 there exists r0 = r0(ε) ∈ (0, 1) such that, for all r ∈ (0, r0),

(129)L((RN \ E) ∩B(x0, r))

L(B(x0, r))< ε.

Assume by contradiction that u 6≡ 0 in Ω. Theorem 1.2 implies that

(130) r−γu(x0 + r(x − x0)) → |x− x0|γψ(0,

x− x0|x− x0|

)as r → 0+,

in C1,α(B1(x0, 1)), where γ = −N−2s2 +

√(N−2s

2 )2 + µk0 and µk0 = µk0(0) > 0 is an eigenvalue of

problem (10) with λ = 0 and ψ is a corresponding eigenfunction. Since u ≡ 0 in E, by (129) wehave

∫

B(x0,r)

u2(x) dx =

∫

(RN\E)∩B(x0,r)

u2(x) dx

6

(∫

(RN\E)∩B(x0,r)

|u(x)|2∗(s)dx)2/2∗(s)

|L((RN \ E) ∩B(x0, r))|2∗(s)−22∗(s)

< ε2∗(s)−22∗(s) |L(B(x0, r))|

2∗(s)−22∗(s)

(∫

(RN\E)∩B(x0,r)

|u(x)|2∗(s)dx)2/2∗(s)

for all r ∈ (0, r0), i.e. letting ur(x) := r−γu(x0 + r(x − x0)),

∫

B(x0,1)

|ur(x)|2dx <(ωN−1

N

)2∗(s)−22∗(s)

ε2∗(s)−22∗(s)

(∫

B(x0,1)

|ur(x)|2∗(s)dx)2/2∗(s)

for all r ∈ (0, r0). Letting r → 0+, from (130) we have that∫

B(x0,1)

|x− x0|2γψ2(0, x−x0

|x−x0|)dx

6

(ωN−1

N

)2∗(s)−22∗(s)

ε2∗(s)−22∗(s)

(∫

B(x0,1)

|x− x0|2∗(s)γψ2∗(s)

(0, x−x0

|x−x0|)dx

)2/2∗(s)

which yields a contradiction as ε→ 0+.

Acknowledgements. The authors would like to thank Prof. Enrico Valdinoci for his interestand for taking their attention to reference [6] and the problem of unique continuation for sets ofpositive measure.


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M.M. FallAfrican Institute for Mathematical Sciences (A.I.M.S.) of Senegal,KM 2, Route de Joal, AIMS-SenegalB.P. 1418. Mbour, Senegal.

E-mail address: [email protected].

V. FelliUniversita di Milano Bicocca,Dipartimento di Matematica e Applicazioni,Via Cozzi 53, 20125 Milano, Italy.

E-mail address: [email protected].

Unique Continuation Property and Local Asymptotics of Solutions to Fractional Elliptic Equations

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